Ramakrishnan

Ramakrishnan Vaidyanathan

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7 years, 74 days

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With twenty years of Industrial experience and twenty years of teaching experience, I am now as retired Professor, using Maple to teach mathematics subject for students studying X to XII standards. Published XII Mathematics books.

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These are answers submitted by Ramakrishnan

Right click the item you want. A display will come with two options as follows

add to facorite palette

Remove from facorite palette

Left click the one desire.

If you want to gack to original click undo in the main  tool bar.

Hope it is useful.

Though i do not know much about the details of the problem, the doc gives an answer if we remove the ri and give ro only in the int fn.

I attach the doc. Hope it gives some insight into it. ro is a fn of ri which ia where there may be a difficulty. If ri and ro are independently specified, it may be easy for maple!.

I may be wrong. I only attempted to remove the error message.

Hope it is useful.

Ramakrishnan VThreePDEfor_site.mw
 

J := f1(r, z)*f2(r, z)*f3(r, z)

`fθ` := f2(r, z)*f3(r, z)/fz

m0 := 1; k0 := 1; k1 := 1; K2 := 1; b := Pi; Ri := 0.3e-2; ph := 0.3e-1; k := 1; l := .9; fz := 1.2; ri := f1(r, z)*f2(r, z)/fz

``

g := r*ph/l

sy := (1/2)*arctan(2*g*fz^2/(-fz^2+g^2+`fθ`^2-1))

I4 := (f2(r, z)^2*cos(sy)^2+f3(r, z)^2*sin(sy)^2-(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2))*cos(b)^2+.5*f2(r, z)*f3(r, z)*(f2(r, z)^2-f3(r, z)^2)*sin(2*sy)*sin(2*b)/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)+((f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)^2+(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2))*sin(b)^2

I6 := (f2(r, z)^2*cos(sy)^2+f3(r, z)^2*sin(sy)^2-(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2))*cos(b)^2-.5*f2(r, z)*f3(r, z)*(f2(r, z)^2-f3(r, z)^2)*sin(2*sy)*sin(2*b)/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)+((f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)^2+(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2))*sin(b)^2

L := (I4-1)*exp(K2*(I4-1)^2-1)

N := (I6-1)*exp(K2*(I6-1)^2-1)

P23 := -f2(r, z)*f3(r, z)*sin(b)*cos(b)+(.5*(f2(r, z)^2-f3(r, z)^2))*sin(sy)^2*sin(b)^2

P32 := -f2(r, z)*f3(r, z)*sin(b)*cos(b)+(.5*(f2(r, z)^2-f3(r, z)^2))*sin(sy)^2*sin(b)^2

P33 := (f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)*sin(b)^2

P22 := ((f2(r, z)^2*cos(sy)^2+f3(r, z)^2*sin(sy)^2)^2-(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2)*cos(b)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)-f2(r, z)*f3(r, z)*(f2(r, z)^2-f3(r, z)^2)*sin(2*sy)*sin(b)*cos(b)/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)+(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2*sin(b)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)

S23 := f2(r, z)*f3(r, z)*sin(b)*cos(b)+(.5*(f2(r, z)^2-f3(r, z)^2))*sin(sy)^2*sin(b)^2

S32 := f2(r, z)*f3(r, z)*sin(b)*cos(b)+(.5*(f2(r, z)^2-f3(r, z)^2))*sin(sy)^2*sin(b)^2

S33 := (f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)*sin(b)^2

S22 := ((f2(r, z)^2*cos(sy)^2+f3(r, z)^2*sin(sy)^2)^2-(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2)*cos(b)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)+f2(r, z)*f3(r, z)*(f2(r, z)^2-f3(r, z)^2)*sin(2*sy)*sin(b)*cos(b)/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)+(.25*(f2(r, z)^2-f3(r, z)^2))*sin(2*sy)^2*sin(b)^2/(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2)

Trr := k0*(J-1)+(1/3)*m0*(2*f1(r, z)^2-f2(r, z)^2-f3(r, z)^2)/J^(5/3)

`Tθθ` := k0*(J-1)+m0*(f2(r, z)^2*cos(sy)^2+f3(r, z)^2*sin(sy)^2-(1/3)*f1(r, z)^2-(1/3)*f2(r, z)^2-(1/3)*f3(r, z)^2)/J^(5/3)+2*k1*(L*S22+N*P22)/J

`Tθz` := m0*(f2(r, z)^2-f3(r, z)^2)*sin(sy)*cos(sy)/J^(5/3)+2*k1*(L*S23+N*P23)/J

`Tzθ` := m0*(f2(r, z)^2-f3(r, z)^2)*sin(sy)*cos(sy)/J^(5/3)+2*k1*(L*S23+N*P23)/J

Tzz := k0*(J-1)+m0*(f2(r, z)^2*sin(sy)^2+f3(r, z)^2*cos(sy)^2-(1/3)*f1(r, z)^2-(1/3)*f2(r, z)^2-(1/3)*f3(r, z)^2)/J^(5/3)+2*k1*(L*S33+N*P33)/J

`Trθ` := 0

`Tθr` := 0

Trz := 0

Tzr := 0

NULLpartialequations:

Typesetting:-mparsed()

(1)

equ(1) := r*(diff(Trr, r))+Trr-`Tθθ`

equ(2) := diff(`Tθz`, z)

equ(3) := diff(Tzz, z)

# BOUNDARY CONDITIONS

"Trr(ro,z):=0: ro=sqrt((f1(r,z)*f2(r,z)*f3(r,z))+(ri)^(2)):"

int((`Tθθ`-Trr)/r, r) := 13.3*10^3

2*Pi*(int(Tzz*r, r, ro))-13.3*Pi*ri^2*10^3

2*Pi*(int((f1(r, z)*f2(r, z)*f3(r, z)-1+(f2(r, z)^2*sin((1/2)*arctan(0.9600000000e-1*r/(-2.44+0.1111111111e-2*r^2+.6944444444*f2(r, z)^2*f3(r, z)^2)))^2+f3(r, z)^2*cos((1/2)*arctan(0.9600000000e-1*r/(-2.44+0.1111111111e-2*r^2+.6944444444*f2(r, z)^2*f3(r, z)^2)))^2-(1/3)*f1(r, z)^2-(1/3)*f2(r, z)^2-(1/3)*f3(r, z)^2)/(f1(r, z)*f2(r, z)*f3(r, z))^(5/3))*r, r))*ro-29016.09881*f1(r, z)^2*f2(r, z)^2

(2)

int(`Tθz`*r^2, r, ri, ro) := 10

10

(3)

NULL


 

Download ThreePDEfor_site.mw

 

@vv 

I thought of obtaining the answers by eval(eq,[0.1,6]) and plot builder commands.

I get the answer as x = -0.4 to 0.1 and y >6.2 to 6 (close to y axis) are the range for eq >0. The rest are all negative field for eq. The conclusion is left to you from the graph.

I attach the doc for reference. where in i have also found the values of eq for sample coordinates.
 

0.1562455469e-15*5^(1/3)*(-0.1579500000e25*5^(1/6)+0.2819757589e24+0.2145450400e30*exp(-3.996000000*5^(2/3)-.2338293009))

 

 

(1)

eval(0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(.1+0.15e-1*y^1.2)), [y = .1])

-83987931.15

(2)

eval(0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(.1+0.15e-1*y^1.2)), [y = -.1])

-0.2521917484e211-0.4425194516e211*I

(3)

eval((1)",["

-0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(-.1+0.15e-1*y^1.2))

(4)

eval(-0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(-.1+0.15e-1*y^1.2)), [y = -.1])

-0.2531926553e211-0.4330042579e211*I

(5)

eval(-0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(-.1+0.15e-1*y^1.2)), [y = .1])

-85592905.26

(6)
 

 

0.8620689655e-15*(-0.3110000000e24*x^2*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(x*y^(2/3)*(x+0.15e-1*y^1.2))

(7)

eval(0.8620689655e-15*(-0.3110000000e24*x^2*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(x*y^(2/3)*(x+0.15e-1*y^1.2)), [x = .1, y = 7])

0.4830348040e-14*7^(1/3)*(-0.1158892000e24*7^(1/6)+0.3704300338e25+0.2145450400e30*exp(-2.854285714*7^(2/3)-.4583054297))

(8)

evalf[5](0.4830348040e-14*7^(1/3)*(-0.1158892000e24*7^(1/6)+0.3704300338e25+0.2145450400e30*exp(-2.854285714*7^(2/3)-.4583054297)))

0.69228e11

(9)
  solveee_vrk.mw
 

restart:

with(SolveTools[Inequality]):

eq:=1/(x*y^(2/3))*8.620689655172415*10^(-16)*(-3.11*10^23*x^2*y^(7/6)-3.92*10^19*y^(25/6)+2.14545039999999*10^29*(0.0108*exp(-45.07/y)+exp(-19.98/y^(1/3)-0.00935317203476387*y^2)))/(x+0.015*y^(1.2));

eval(eq,[x=0.1]);

0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(.1+0.15e-1*y^1.2))

(1)

eval(0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(.1+0.15e-1*y^1.2)), [y = .1])

-83987931.15

(2)

eval(0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(.1+0.15e-1*y^1.2)), [y = -.1])

-0.2521917484e211-0.4425194516e211*I

(3)

eval((1)",["

-0.8620689655e-14*(-0.3110000000e22*y^(7/6)-0.3920000000e20*y^(25/6)+0.2317086432e28*exp(-45.07/y)+0.2145450400e30*exp(-19.98/y^(1/3)-0.935317203476387e-2*y^2))/(y^(2/3)*(-.1+0.15e-1*y^1.2))

(4)

plots:-implicitplot(.8620689655e-15/x/y^(2/3)*(-.3110000000e24*x^2*y^(7/6)-.3920000000e20*y^(25/6)+.2317086432e28*exp(-45.07/y)+.2145450400e30*exp(-19.98/y^(1/3)-.935317203476387e-2*y^2))/(x+.15e-1*y^1.2),
             x = -10.0 .. 10.0, y = -10.0 .. 10.0)

 

``


Download solveee_vrk.mw

 

 

 

 

   
 

 

 

 

 

 


 

Download solveee_vrk.mw

 

The problem i understand was you first check whether an answer is possible and then you start finding an answer.

Here concavity checks the possibility and equating the differentials give the answer.

I am now surprised that there is a method which gives the answer directly(!). It proves that maximum exists and hence concavity as well exists.

I checked the maple doc  for solving the four differentials with respect to the four variables and am yet to get the evaluation completed (half an hour passed). If what was found is correct by global optimisation package, then, maple can as well use this techniquefor solving atleast this particular case of four equations with four variables.

Attached maple struggling document in maple. (or may be it may take more than I can wait with patience for an answer)

We can not complain that maple has not solved it. It takes longer time and we lose patience and concavity_proof_question_(1).mw
 

sol1 := TP1 = (1/6)*(-(3*((4/3)*W^beta*m*E^(-beta+1)-(2/3)*alpha*(-m+T)^2*(beta-1)*(b*p-a)*W^beta+m*((E-W)*beta-E-(1/3)*W)))*h*sqrt((2*E^(-beta+1)*m-2*m*W^(-beta+1)+alpha*(-m+T)^2*(beta-1)*(-b*p+a))/(alpha*(beta-1)*(-b*p+a)))-6*W^beta*m*(-h*m-c+p)*E^(-beta+1)+3*alpha*(beta-1)*(b*p-a)*(-(2/3)*h*m^3+2*h*T*m^2-2*T*((1/2)*h*T+p-c)*m+T^2*(p-c))*W^beta-(6*(-(1/2*((E-W)*beta-E-W))*h*m+(1/2*((W+E)*h+2*u*W))*(beta-1)*T+((E-W)*p-s*E+W*c+o)*beta-E*p+s*E-o))*m)/((beta-1)*m)

TP1 = (1/6)*(-(4*W^beta*m*E^(-beta+1)-2*alpha*(-m+T)^2*(beta-1)*(b*p-a)*W^beta+3*m*((E-W)*beta-E-(1/3)*W))*h*((2*E^(-beta+1)*m-2*m*W^(-beta+1)+alpha*(-m+T)^2*(beta-1)*(-b*p+a))/(alpha*(beta-1)*(-b*p+a)))^(1/2)-6*W^beta*m*(-h*m-c+p)*E^(-beta+1)+3*alpha*(beta-1)*(b*p-a)*(-(2/3)*h*m^3+2*h*T*m^2-2*T*((1/2)*h*T+p-c)*m+T^2*(p-c))*W^beta-(-6*((1/2)*(E-W)*beta-(1/2)*E-(1/2)*W)*h*m+6*((1/2)*(W+E)*h+u*W)*(beta-1)*T+6*((E-W)*p-s*E+W*c+o)*beta-6*E*p+6*s*E-6*o)*m)/((beta-1)*m)

(1)

NULL

para1 := [alpha = 50, beta = .7, c = 20, h = 4, m = .4, o = 10, s = 10, u = 5, a = 7, b = .15]

[alpha = 50, beta = .7, c = 20, h = 4, m = .4, o = 10, s = 10, u = 5, a = 7, b = .15]

(2)

sol2 := eval(sol1, para1)

TP1 = 5.555555555*(1.6*W^.7*E^.3+30.0*(-.4+T)^2*(.15*p-7)*W^.7-.36*E-1.240000000*W)*(-0.6666666667e-1*(.8*E^.3-.8*W^.3-15.0*(-.4+T)^2*(-.15*p+7))/(-.15*p+7))^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3+62.49999998*(.15*p-7)*(-.1706666667+1.28*T-.8*T*(2*T+p-20)+T^2*(p-20))*W^.7+10.80000000*E+51.20000000*W-.9999999998*(7*W+2*E)*T+2.333333333*(E-W)*p-9.999999998-3.333333333*E*p

(3)

NULL

NULL

sol3 := TP2 = (1/6)*(-(3*((4/3)*W^beta*m*E^(-beta+1)+(2/3)*alpha*W^beta*a*(-m+T)^2*(beta-1)*p^(-b)+((E-W)*beta-E-(1/3)*W)*m))*h*sqrt((2*E^(-beta+1)*p^b*m-2*p^b*W^(-beta+1)*m+a*alpha*(-m+T)^2*(beta-1))/(a*alpha*(beta-1)))-6*W^beta*m*(-h*m-c+p)*E^(-beta+1)-3*alpha*W^beta*T*a*(T-2*m)*(beta-1)*p^(-b+1)+3*alpha*W^beta*a*(beta-1)*((2/3)*h*m^3-2*T*h*m^2+T*(T*h-2*c)*m+T^2*c)*p^(-b)-(6*(-(1/2)*h*((E-W)*beta-E-W)*m+(1/2)*T*(beta-1)*(W+E)*h+(u*W*T+(p-s)*E+(-p+c)*W+o)*beta-u*W*T+(-p+s)*E-o))*m)/(m*(beta-1))

TP2 = (1/6)*(-(4*W^beta*m*E^(-beta+1)+2*alpha*W^beta*a*(-m+T)^2*(beta-1)*p^(-b)+3*m*((E-W)*beta-E-(1/3)*W))*h*((2*E^(-beta+1)*p^b*m-2*p^b*W^(-beta+1)*m+a*alpha*(-m+T)^2*(beta-1))/(a*alpha*(beta-1)))^(1/2)-6*W^beta*m*(-h*m-c+p)*E^(-beta+1)-3*alpha*W^beta*T*a*(T-2*m)*(beta-1)*p^(-b+1)+3*alpha*W^beta*a*(beta-1)*((2/3)*h*m^3-2*h*T*m^2+T*(T*h-2*c)*m+T^2*c)*p^(-b)-(-3*h*((E-W)*beta-E-W)*m+3*T*(beta-1)*(W+E)*h+6*(u*W*T+(p-s)*E+(-p+c)*W+o)*beta-6*u*W*T+6*(-p+s)*E-6*o)*m)/(m*(beta-1))

(4)

NULL

para2 := [alpha = 50, beta = .7, c = 20, h = 4, m = .4, o = 10, s = 10, u = 5, a = 100, b = 1.2]

[alpha = 50, beta = .7, c = 20, h = 4, m = .4, o = 10, s = 10, u = 5, a = 100, b = 1.2]

(5)

sol4 := eval(sol3, para2)

TP2 = 5.555555555*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3-6250.000000*W^.7*T*(T-.8)/p^.2+6250.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^1.2+.8000000000*E+4.533333333*W-2.000000000*T*(W+E)-5.000000000*W*T+2.333333333*(p-10)*E+2.333333333*(-p+20)*W-10.00000000+3.333333333*(-p+10)*E

(6)

diff(TP2 = 5.555555555*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3-6250.000000*W^.7*T*(T-.8)/p^.2+6250.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^1.2+.8000000000*E+4.533333333*W-2.000000000*T*(W+E)-5.000000000*W*T+2.333333333*(p-10)*E+2.333333333*(-p+20)*W-10.00000000+3.333333333*(-p+10)*E, p)

0 = 20000.00000*W^.7*(-.4+T)^2*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)/p^2.2+2.777777778*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.6400000001e-3*E^.3*p^.2+0.6400000001e-3*p^.2*W^.3)/(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*E^.3+1250.000000*W^.7*T*(T-.8)/p^1.2-7500.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^2.2-1.000000000*E-2.333333333*W

(7)

diff(TP2 = 5.555555555*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3-6250.000000*W^.7*T*(T-.8)/p^.2+6250.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^1.2+.8000000000*E+4.533333333*W-2.000000000*T*(W+E)-5.000000000*W*T+2.333333333*(p-10)*E+2.333333333*(-p+20)*W-10.00000000+3.333333333*(-p+10)*E, W)

0 = 5.555555555*(1.12*E^.3/W^.3-2100.00*(-.4+T)^2/(W^.3*p^1.2)-1.240000000)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+0.4444444444e-3*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*p^1.2/((-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)*W^.7)+2.333333333*(p-21.6)*E^.3/W^.3-4375.000000*T*(T-.8)/(W^.3*p^.2)+4375.000000*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/(W^.3*p^1.2)+51.19999999-7.000000000*T-2.333333333*p

(8)

diff(TP2 = 5.555555555*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3-6250.000000*W^.7*T*(T-.8)/p^.2+6250.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^1.2+.8000000000*E+4.533333333*W-2.000000000*T*(W+E)-5.000000000*W*T+2.333333333*(p-10)*E+2.333333333*(-p+20)*W-10.00000000+3.333333333*(-p+10)*E, T)

0 = -33333.33333*W^.7*(-.4+T)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)/p^1.2+2.777777778*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-.8000000000+2.000000000*T)/(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)-6250.000000*W^.7*(T-.8)/p^.2-6250.000000*W^.7*T/p^.2+6250.000000*W^.7*(-17.28+43.2*T)/p^1.2-7.000000000*W-2.000000000*E

(9)

diff(TP2 = 5.555555555*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*(p-21.6)*E^.3-6250.000000*W^.7*T*(T-.8)/p^.2+6250.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^1.2+.8000000000*E+4.533333333*W-2.000000000*T*(W+E)-5.000000000*W*T+2.333333333*(p-10)*E+2.333333333*(-p+20)*W-10.00000000+3.333333333*(-p+10)*E, E)

0 = 5.555555555*(.48*W^.7/E^.7-.36)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)-0.4444444444e-3*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*p^1.2/((-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)*E^.7)+.9999999999*W^.7*(p-21.6)/E^.7+10.80000000-2.000000000*T-1.000000000*p

(10)

solve({0 = 5.555555555*(.48*W^.7/E^.7-.36)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)-0.4444444444e-3*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*p^1.2/((-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)*E^.7)+.9999999999*W^.7*(p-21.6)/E^.7+10.80000000-2.000000000*T-1.000000000*p, 0 = -33333.33333*W^.7*(-.4+T)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)/p^1.2+2.777777778*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-.8000000000+2.000000000*T)/(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)-6250.000000*W^.7*(T-.8)/p^.2-6250.000000*W^.7*T/p^.2+6250.000000*W^.7*(-17.28+43.2*T)/p^1.2-7.000000000*W-2.000000000*E, 0 = 20000.00000*W^.7*(-.4+T)^2*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)/p^2.2+2.777777778*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*(-0.6400000001e-3*E^.3*p^.2+0.6400000001e-3*p^.2*W^.3)/(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+3.333333333*W^.7*E^.3+1250.000000*W^.7*T*(T-.8)/p^1.2-7500.000000*W^.7*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/p^2.2-1.000000000*E-2.333333333*W, 0 = 5.555555555*(1.12*E^.3/W^.3-2100.00*(-.4+T)^2/(W^.3*p^1.2)-1.240000000)*(-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)+0.4444444444e-3*(1.6*W^.7*E^.3-3000.0*W^.7*(-.4+T)^2/p^1.2-.36*E-1.240000000*W)*p^1.2/((-0.5333333334e-3*E^.3*p^1.2+0.5333333334e-3*p^1.2*W^.3+1.000000000*(-.4+T)^2)^(1/2)*W^.7)+2.333333333*(p-21.6)*E^.3/W^.3-4375.000000*T*(T-.8)/(W^.3*p^.2)+4375.000000*(.1706666667-1.28*T+.4*T*(4*T-40)+20*T^2)/(W^.3*p^1.2)+51.19999999-7.000000000*T-2.333333333*p})

Warning,  computation interrupted

 

with(Student[VectorCalculus])

Hess_TP1 := Hessian(rhs(sol2), [T, E, W, p])

Hess_TP2 := Hessian(rhs(sol4), [T, E, W, p])

NULL

Warning,  computation interrupted

 

IsDefinite(Hess_TP2, 'query' = 'negative_semidefinite')


 

Download concavity_proof_question_(1).mw

interrupt to cancel evaluation.

Cheers.

Ramakrishnan V

There are two methods and the second one is comfortable in this case. 

The methods are (in terms of conditions)

(1) If there exists a positive number r such that for every independent variable x in (a − ra) we have f′(x) ≥ 0, and for every x in (aa + r) we have f′(x) ≤ 0, then f has a local maximum at a.

(2) If f"<0,  then f has a local fmaximum at x (independent variable value)

Hence find second derivative of both TP1 and TP2 in terms of the four independent variables  T,E,W,p separately and if all of them are negative, then you have an optimum f value. Else only those variable that give f"<0 behave such as to give you an optimum f.

Hope I have not confused your already somewhat clear mind!

Cheers.

Ramakrishnan v

Name the plots and use display plots command.

Also in the following command, sol1b_1 is not properly spelled.

gr1c:=multiple(plot[soln1a,soln1b1],dom1,color=[blue,purple])

with(plots); with(DEtools); K := 9; deG := diff(theta(t), t, t)+mu*(diff(theta(t), t))+K*sin(theta(t)) = 0; deL := diff(theta(t), t, t)+mu*(diff(theta(t), t))+K*theta(t) = 0; Iv := theta(0) = .75, (D(theta))(0) = 2.0; dom1 := t = 0 .. 10; soln1a := dsolve({Iv, eval(deL, mu = 0)}); soln1b_1 := dsolve({Iv, eval(deL, mu = 1)}); gr1c := multiple(plot[soln1a, soln1b_1], dom1, color = [blue, purple])

theta(t) = (19/140)*35^(1/2)*exp(-(1/2)*t)*sin((1/2)*35^(1/2)*t)+(3/4)*exp(-(1/2)*t)*cos((1/2)*35^(1/2)*t)

(1)

gr1b1 := plot(rhs(soln1b_1), dom1, color = purple)

 

gr1a1 := plot(rhs(soln1a), dom1, color = purple)

 

plots:-display(gr1a1, gr1b1)

 

``


 

Download MultiplePlot_mapleprime_28062017.mwMultiplePlot_mapleprime_28062017.mw

@Rouben Rostamian  

Hope the following commands also would be ok.

Ramakrishnan V
 

restart:

data2D := [[1, 2], [3, 4], [5, 10], [20, 30], [40, 50]]

[[1, 2], [3, 4], [5, 10], [20, 30], [40, 50]]

(1)

NULL

f := Spline(data2D, v, degree = 1)

piecewise(v < 3, 1+v, v < 5, -5+3*v, v < 20, 10/3+(4/3)*v, 10+v)

(2)

"->""->"

 

NULL


 

Download spline_for_mapleprime_26062017.mwspline_for_mapleprime_26062017.mw

If I am correct in understanding that there are 12 equations with as many  variables, I just put the equations and variables, the Maple gives the answers. However few of them are interdependent and can be simplified (of course  we should solve once more the resultant variables in equation form). 

Hope the answers satisfy you. Ramakrishnan vsolving12Equns.mw
 

restart

b[1](2)*a[1](2, 1) = 1/2, b[1](2)*a[1](2, 1)^2 = 1/3

b[1](2)*a[1](2, 1) = 1/2, b[1](2)*a[1](2, 1)^2 = 1/3

(1)

b[1](2)*a[2](2, 1) = 1/2

b[1](2)*a[2](2, 1) = 1/2

(2)

b[1](2)*a[2](2, 1)^2 = 1/3

b[1](2)*a[2](2, 1)^2 = 1/3

(3)

b[2](2)*a[1](2, 1) = 1/2

b[2](2)*a[1](2, 1) = 1/2

(4)

b[2](2)*a[2](2, 1) = 1/2

b[2](2)*a[2](2, 1) = 1/2

(5)

b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[1](2, 1) = 1/3

b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[1](2, 1) = 1/3

(6)

b[1](1)*(a[3](1, 1)+a[3](1, 2))+b[1](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2

b[1](1)*(a[3](1, 1)+a[3](1, 2))+b[1](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2

(7)

b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[2](2, 1) = 1/3

b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[2](2, 1) = 1/3

(8)

b[1](2)*a[2](2, 1)*a[1](2, 1) = 1/3

b[1](2)*a[2](2, 1)*a[1](2, 1) = 1/3

(9)

b[1](1)*(a[3](1, 1)+a[3](1, 2))^2+b[1](2)*(a[3](2, 1)+a[3](2, 2))^2 = 1/3

b[1](1)*(a[3](1, 1)+a[3](1, 2))^2+b[1](2)*(a[3](2, 1)+a[3](2, 2))^2 = 1/3

(10)

b[2](1)*(a[3](1, 1)+a[3](1, 2))+b[2](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2

b[2](1)*(a[3](1, 1)+a[3](1, 2))+b[2](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2

(11)

``

b[1](1)+b[1](2) = 1, b[2](1)+b[2](2) = 1

b[1](1)+b[1](2) = 1, b[2](1)+b[2](2) = 1

(12)

variables := {a[1](2, 1), a[2](2, 1), a[3](1, 1), a[3](1, 2), a[3](2, 1), a[3](2, 2), b[1](1), b[1](2), b[2](1), b[2](2), b[3](1), b[3](2)}

{a[1](2, 1), a[2](2, 1), a[3](1, 1), a[3](1, 2), a[3](2, 1), a[3](2, 2), b[1](1), b[1](2), b[2](1), b[2](2), b[3](1), b[3](2)}

(13)

solve({b[1](2)*a[1](2, 1) = 1/2, b[1](2)*a[1](2, 1)^2 = 1/3, b[1](2)*a[2](2, 1) = 1/2, b[1](2)*a[2](2, 1)^2 = 1/3, b[2](2)*a[1](2, 1) = 1/2, b[2](2)*a[2](2, 1) = 1/2, b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[1](2, 1) = 1/3, b[1](2)*(a[3](2, 1)+a[3](2, 2))*a[2](2, 1) = 1/3, b[1](2)*a[2](2, 1)*a[1](2, 1) = 1/3, b[1](1)*(a[3](1, 1)+a[3](1, 2))+b[1](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2, b[1](1)*(a[3](1, 1)+a[3](1, 2))^2+b[1](2)*(a[3](2, 1)+a[3](2, 2))^2 = 1/3, b[2](1)*(a[3](1, 1)+a[3](1, 2))+b[2](2)*(a[3](2, 1)+a[3](2, 2)) = 1/2, b[1](1)+b[1](2) = 1, b[2](1)+b[2](2) = 1}, {a[1](2, 1), a[2](2, 1), a[3](1, 1), a[3](1, 2), a[3](2, 1), a[3](2, 2), b[1](1), b[1](2), b[2](1), b[2](2), b[3](1), b[3](2)})

{a[1](2, 1) = 2/3, a[2](2, 1) = 2/3, a[3](1, 1) = a[3](1, 1), a[3](1, 2) = -a[3](1, 1), a[3](2, 1) = a[3](2, 1), a[3](2, 2) = -a[3](2, 1)+2/3, b[1](1) = 1/4, b[1](2) = 3/4, b[2](1) = 1/4, b[2](2) = 3/4, b[3](1) = b[3](1), b[3](2) = b[3](2)}

(14)

``


 

Download solving12Equns.mw

 

Maple2016 does perfectly well this problem.

``

proc (t) options operator, arrow; (2*t-1)*cos*sqrt(3*(2*t-1)^2+6)/sqrt(3*(2*t-1)^2+6) end proc

(1)

"(->)"

proc (t) options operator, arrow; 2*cos end proc

(2)

h = (2*t-1)*cos(sqrt(3*(2*t-1)^2+6))/sqrt(3*(2*t-1)^2+6)

h = (2*t-1)*cos((3*(2*t-1)^2+6)^(1/2))/(3*(2*t-1)^2+6)^(1/2)

(3)

"(->)"

h*t = (1/6)*sin((12*t^2-12*t+9)^(1/2))

(4)

"(->)"

h = (1/12)*(24*t-12)*cos((12*t^2-12*t+9)^(1/2))/(12*t^2-12*t+9)^(1/2)

(5)

"(=)"

h = (2*t-1)*cos((12*t^2-12*t+9)^(1/2))/(12*t^2-12*t+9)^(1/2)

(6)

````

``

``

3*(2*t-1)^2+6

3*(2*t-1)^2+6

(7)

"(=)"

12*t^2-12*t+9

(8)

``

``

``

``


Download AnswerTo_mapleprime.mw

Ramakrishnan V

rukmini_ramki@hotmail.com


Hope this is acceptable.

a := 0:

y1 := sin(x+a);

sin(x)

(1)

y2 := sin(x+b)

sin(x+1)

(2)

``

F := `<,>`(y1, y2)

F := Vector(2, {(1) = sin(x), (2) = sin(x+1)})

(3)

plot(convert(F, list), x = 0 .. 10,  color = [red, black], scaling = constrained);

 

``


Download ThisWayDrawTheGraph.mwThisWayDrawTheGraph.mw

Ramakrishnan V

rukmini_ramki@hotmail.com


The differential equation is

5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0

5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0

(1)

with boundary conditions

(D(y))(0) = a, y(0) = b

(D(y))(0) = a, y(0) = b

(2)

Isolating  diff(y(x), x) and substituting x = 0, we get y'(0) interms of 'b' and thus 'a' is eliminated.

5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0 = 5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0"(->)"diff(y(x), x) = -(1/5)*y(x)-(2/5)*a*x^3-(3/5)*b"(->)"eval(diff(y(x), x), {x = 0}) = -(1/5)*y(0)-(3/5)*b"(->)"eval(diff(y(x), x), {x = 0}) = -(4/5)*b``

Result  now is (D(y))(0) = -(4/5)*b

Substituting the value of (D(y))(0) in terms of b and solving for y(x) we get an additional constant _C1. Applying bounadry condition y(0) = b, we get _C1 in terms of b.

5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0 = 5*(diff(y(x), x))+y(x)+2*a*x^3+3*b = 0"(->)"5*(diff(y(x), x))+y(x)-(8/5)*b*x^3+3*b = 0"(->)"y(x) = -24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+exp(-(1/5)*x)*_C1``

y(x) = -24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+exp(-(1/5)*x)*_C1"(->)"b = -1203*b+_C1"(->)"[[_C1 = 1204*b]]NULL

Substituting the value of _C1 we get the final solution as

y(x) = 24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+1204*b*exp(-(1/5)*x)

 

Verification

y(x) = 24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+1204*b*exp(-(1/5)*x) = y(x) = 24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+1204*b*exp(-(1/5)*x)"(->)"diff(y(x), x) = 48*b*x+(24/5)*b*x^2+240*b-(1204/5)*b*exp(-(1/5)*x)"(->)"eval(diff(y(x), x), {x = 0}) = -(4/5)*b``

y(x) = 24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+1204*b*e^(-(1/5)*x)"(->)"b = b

The solution is valid for any value of b including unit value. Substituting b = 1, the solution is

y(x) = 24*b*x^2+(8/5)*b*x^3+240*b*x-1203*b+1204*b*e^(-(1/5)*x)"(->)"y(x) = 24*x^2+(8/5)*x^3+240*x-1203+1204*e^(-(1/5)*x)``

Correct Answer in my opinion is

y(x) = 24*x^2+(8/5)*x^3+240*x-1203+1204*e^(-(1/5)*x)

 

NULL


Download solving_differential_equation_22Dec2015.mwsolving_differential_equation_22Dec2015.mw

I hope you are satisfied with my solution in attached document and reproduced below. Maple experts may obtain the same solution in easy commands.

Thanks for the problem.

 

Ramakrishnan V

rukmini_ramki@hotmail.com


equn := arctan(x)+arctan((1/2)*x) = (1/2)*Pi

``

solve({equn}, {x})

{x = (1/3)*6^(1/2)*3^(1/2)}

(1)

"(->)"

{x = 2^(1/2)}

(2)

Aliter substitute the equation itself in the command as follows

solve({arctan(x)+arctan((1/2)*x) = (1/2)*Pi}, {x})

{x = (1/3)*6^(1/2)*3^(1/2)}

(3)

"(->)"

{x = 2^(1/2)}

(4)

``


Download solutiontosolve_equation.mwsolutiontosolve_equation.mw

Ramakrishnan V

rukmini_ramki@hotmail.com


restart

with(Statistics):

``

Example 1

XYZ := Matrix(10, 3, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 4, (2, 1) = 2, (2, 2) = 3, (2, 3) = 5, (3, 1) = 3, (3, 2) = 4, (3, 3) = 6, (4, 1) = 4, (4, 2) = 5, (4, 3) = 8, (5, 1) = 5, (5, 2) = 6, (5, 3) = 10, (6, 1) = 6, (6, 2) = 7, (6, 3) = 13, (7, 1) = 7, (7, 2) = 8, (7, 3) = 15, (8, 1) = 8, (8, 2) = 9, (8, 3) = 18, (9, 1) = 9, (9, 2) = 10, (9, 3) = 20, (10, 1) = 10, (10, 2) = 11, (10, 3) = 21})

Matrix([[1, 2, 4], [2, 3, 5], [3, 4, 6], [4, 5, 8], [5, 6, 10], [6, 7, 13], [7, 8, 15], [8, 9, 18], [9, 10, 20], [10, 11, 21]])

(1)

plots[pointplot3d](XYZ, glossiness = 1.0, title = "velocity ay space", caption = "plot u vs x,y", labeldirections = [HORIZONTAL, HORIZONTAL, VERTICAL], view = [0 .. 10, 0 .. 10, 0 .. 25], labels = [x, y, u]);

 

NULL

NULL

``

``

Example 2

A := Matrix(4, 3, {(1, 1) = 1, (1, 2) = 1, (1, 3) = 1, (2, 1) = 2, (2, 2) = 2, (2, 3) = 4, (3, 1) = 3, (3, 2) = 3, (3, 3) = 9, (4, 1) = 4, (4, 2) = 4, (4, 3) = 16})"->"NULL

plots[pointplot3d](A, glossiness = 1.0, title = "velocity ay space", caption = "plot u vs x,y", labeldirections = [HORIZONTAL, HORIZONTAL, VERTICAL], view = [0 .. 5, 0 .. 5, 0 .. 20], labels = [x, y, u]);

 

NULL

PQR := Matrix(4, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 2, (2, 2) = 2, (3, 1) = 3, (3, 2) = 9, (4, 1) = 4, (4, 2) = 16})NULL

plots[lineplot](PQR, color = "Black", labeldirections = [HORIZONTAL, VERTICAL], view = [0 .. 5, 0 .. 16])

 

``


Download pointplot3D.mw

I have attempted. Hope it is of use.

Ramakrishnan V

rukmini_ramki@hotmail.com


restart; f1a := -sqrt(N)*omega^(3/2)*sin(theta[2])*cos(`&varphi;`[2])*`&lambda;__b`+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(`&varphi;`[1])*`&lambda;__a`; f1b := sqrt(N)*omega^(3/2)*sin(theta[2])*cos(`&varphi;`[2])*`&lambda;__b`+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(`&varphi;`[1])*`&lambda;__a`

f2 := kappa*omega^2+omega^3:

f3 := -kappa*omega^2+omega^3:

f4 := Omega*N*cos(theta[2])*omega+Omega*N*cos(theta[1])*omega-P__X^2*kappa+P__X^2*omega+P__Y^2*kappa+P__Y^2*omega:

f5 := sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b:

f6 := sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a

N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a

(1)

f7 := (sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2:

term3old := (Omega*N*cos(theta[2])*omega+Omega*N*cos(theta[1])*omega-P__X^2*kappa+P__X^2*omega+P__Y^2*kappa+P__Y^2*omega)/(2*omega):

term3new := f4/(2*omega);

(1/2)*(Omega*N*cos(theta[2])*omega+Omega*N*cos(theta[1])*omega-P__X^2*kappa+P__X^2*omega+P__Y^2*kappa+P__Y^2*omega)/omega

(2)

term3new-term3old;

0

(3)

term4old := (sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(8*omega*(-kappa*omega^2+omega^3)):

term4new := (f5+f6)^2/(8*omega*f3);

(1/8)*(N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(omega*(-kappa*omega^2+omega^3))

(4)

NULL

term4new-term4old;

0

(5)

term5old := (-sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(8*omega*(kappa*omega^2+omega^3)):

term5new := (-f5+f6)^2/(8*omega*f2):

term5new-term5old;

0

(6)

term1new := f2*(Y+(1/2)*f1a/f2)^2/(2*omega):

NULL``

``

``

term1old := (kappa*omega^2+omega^3)*(Y+(1/2)*(-sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(kappa*omega^2+omega^3))^2/(2*omega):

term1new-term1old;

0

(7)

term2new := f3*(X+(1/2)*f1b/f3)^2/(2*omega):

term2old := (-kappa*omega^2+omega^3)*(X+(1/2)*(sqrt(N)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+sqrt(N)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(-kappa*omega^2+omega^3))^2/(2*omega):

term2new-term2old;

0

(8)

NewExpression := term1new+term2new+term3new+term4new+term5new;

(1/2)*(kappa*omega^2+omega^3)*(Y+(1/2)*(-N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(kappa*omega^2+omega^3))^2/omega+(1/2)*(-kappa*omega^2+omega^3)*(X+(1/2)*(N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(-kappa*omega^2+omega^3))^2/omega+(1/2)*(Omega*N*cos(theta[2])*omega+Omega*N*cos(theta[1])*omega-P__X^2*kappa+P__X^2*omega+P__Y^2*kappa+P__Y^2*omega)/omega+(1/8)*(N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(omega*(-kappa*omega^2+omega^3))+(1/8)*(-N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(omega*(kappa*omega^2+omega^3))

(9)

f:

NULL

oldExpression := term1old+term2old+term3old+term4old+term5old:

NewExpression-oldExpression;

0

(10)

``

NewExpression;

(1/2)*(kappa*omega^2+omega^3)*(Y+(1/2)*(-N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(kappa*omega^2+omega^3))^2/omega+(1/2)*(-kappa*omega^2+omega^3)*(X+(1/2)*(N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)/(-kappa*omega^2+omega^3))^2/omega+(1/2)*(Omega*N*cos(theta[2])*omega+Omega*N*cos(theta[1])*omega-P__X^2*kappa+P__X^2*omega+P__Y^2*kappa+P__Y^2*omega)/omega+(1/8)*(N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(omega*(-kappa*omega^2+omega^3))+(1/8)*(-N^(1/2)*omega^(3/2)*sin(theta[2])*cos(varphi[2])*lambda__b+N^(1/2)*omega^(3/2)*sin(theta[1])*cos(varphi[1])*lambda__a)^2/(omega*(kappa*omega^2+omega^3))

(11)

"(=)"

(1/2)*(-omega^(7/2)*(omega-kappa)*(kappa+omega)*(sin(theta[1])*lambda__a*(X+Y)*cos(varphi[1])+lambda__b*sin(theta[2])*cos(varphi[2])*(X-Y))*N^(1/2)-(N*sin(theta[1])^2*omega^2*lambda__a^2*cos(varphi[1])^2+2*N*cos(varphi[1])*cos(varphi[2])*sin(theta[2])*sin(theta[1])*kappa*omega*lambda__a*lambda__b+N*lambda__b^2*sin(theta[2])^2*cos(varphi[2])^2*omega^2-(Omega*N*cos(theta[1])*omega+Omega*N*cos(theta[2])*omega+(X^2+Y^2)*omega^3+(-X^2+Y^2)*kappa*omega^2+(P__X^2+P__Y^2)*omega+(-P__X^2+P__Y^2)*kappa)*(kappa-omega)*(kappa+omega))*omega^2)/(omega^3*(kappa+omega)*(kappa-omega))

(12)

``

NULL

NULL


Download SimplifyTerms.mw

I hope the simplification done by me would help you.

Ramakrishnan V

rukmini_ramki@hotmail.com

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