Question: How factoring two term after substituting solution in ODE?

I already get the same results, but there's something about the factoring process that I encountered for the first time in this ODE. In the paper, it says that G′ satisfies a certain condition, but I’m not sure exactly what that means. Did the author use it for substitution, or did they change (m+F) into another variable and then solve? I’m not exactly sure what approach was taken. Does anyone have any idea or insight into this?

restart

with(PDEtools)

with(LinearAlgebra)

NULL

with(SolveTools)

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

(1)

declare(Omega(x, t)); declare(U(xi)); declare(u(x, y, z, t)); declare(Q(xi))

Omega(x, t)*`will now be displayed as`*Omega

 

U(xi)*`will now be displayed as`*U

 

u(x, y, z, t)*`will now be displayed as`*u

 

Q(xi)*`will now be displayed as`*Q

(2)

tr := {t = tau, x = (-ZETA*c[3]-tau*c[4]-`Υ`*c[2]+xi)/c[1], y = `Υ`, z = ZETA, u(x, y, z, t) = U(xi)}

{t = tau, x = (-Zeta*c[3]-tau*c[4]-`Υ`*c[2]+xi)/c[1], y = `Υ`, z = Zeta, u(x, y, z, t) = U(xi)}

(3)

pde1 := diff(u(x, y, z, t), `$`(x, 3), z)-4*(diff(u(x, y, z, t), x, t))+4*(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), x, z))+2*(diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), z))+3*(diff(u(x, y, z, t), `$`(y, 2))) = 0

diff(diff(diff(diff(u(x, y, z, t), x), x), x), z)-4*(diff(diff(u(x, y, z, t), t), x))+4*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), z))+2*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), z))+3*(diff(diff(u(x, y, z, t), y), y)) = 0

(4)

``

L1 := PDEtools:-dchange(tr, pde1, [xi, `Υ`, ZETA, tau, U])

(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*c[1]^3*c[3]-4*(diff(diff(U(xi), xi), xi))*c[4]*c[1]+6*(diff(U(xi), xi))*c[1]^2*(diff(diff(U(xi), xi), xi))*c[3]+3*(diff(diff(U(xi), xi), xi))*c[2]^2 = 0

(5)

map(int, (diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*c[1]^3*c[3]-4*(diff(diff(U(xi), xi), xi))*c[4]*c[1]+6*(diff(U(xi), xi))*c[1]^2*(diff(diff(U(xi), xi), xi))*c[3]+3*(diff(diff(U(xi), xi), xi))*c[2]^2 = 0, xi)

c[1]^3*c[3]*(diff(diff(diff(U(xi), xi), xi), xi))+3*c[2]^2*(diff(U(xi), xi))-4*c[4]*c[1]*(diff(U(xi), xi))+3*c[1]^2*c[3]*(diff(U(xi), xi))^2 = 0

(6)

ode := %

c[1]^3*c[3]*(diff(diff(diff(U(xi), xi), xi), xi))+3*c[2]^2*(diff(U(xi), xi))-4*c[4]*c[1]*(diff(U(xi), xi))+3*c[1]^2*c[3]*(diff(U(xi), xi))^2 = 0

(7)

F := sum(a[i]*(m+1/(diff(G(xi), xi)))^i, i = -1 .. 1)

a[-1]/(m+1/(diff(G(xi), xi)))+a[0]+a[1]*(m+1/(diff(G(xi), xi)))

(8)

D1 := diff(F, xi)

a[-1]*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^2

(9)

S := diff(G(xi), `$`(xi, 2)) = -(2*m*mu+lambda)*(diff(G(xi), xi))-mu

diff(diff(G(xi), xi), xi) = -(2*m*mu+lambda)*(diff(G(xi), xi))-mu

(10)

E1 := subs(S, D1)

a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(11)

D2 := diff(E1, xi)

2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)-2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)-a[-1]*(2*m*mu+lambda)*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)+2*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^3+a[1]*(2*m*mu+lambda)*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^2

(12)

E2 := subs(S, D2)

2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)-2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)-a[-1]*(2*m*mu+lambda)*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)+2*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/(diff(G(xi), xi))^3+a[1]*(2*m*mu+lambda)*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(13)

D3 := diff(E2, xi)

6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^4*(diff(G(xi), xi))^6)-12*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^5)-6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(2*m*mu+lambda)*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(2*m*mu+lambda)*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)+a[-1]*(2*m*mu+lambda)^2*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^4-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)*(2*m*mu+lambda)*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^3-a[1]*(2*m*mu+lambda)^2*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^2

(14)

E3 := subs(S, D3)

6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^4*(diff(G(xi), xi))^6)-12*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^5)-6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)+a[-1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/(diff(G(xi), xi))^4-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/(diff(G(xi), xi))^3-a[1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(15)

NULL

NULL

K := U(xi) = F

U(xi) = a[-1]/(m+1/(diff(G(xi), xi)))+a[0]+a[1]*(m+1/(diff(G(xi), xi)))

(16)

K1 := diff(U(xi), xi) = E1

diff(U(xi), xi) = a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(17)

K2 := diff(U(xi), `$`(xi, 2)) = E2

diff(diff(U(xi), xi), xi) = 2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)-2*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)-a[-1]*(2*m*mu+lambda)*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)+2*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2/(diff(G(xi), xi))^3+a[1]*(2*m*mu+lambda)*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(18)

K3 := diff(U(xi), `$`(xi, 3)) = E3

diff(diff(diff(U(xi), xi), xi), xi) = 6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^4*(diff(G(xi), xi))^6)-12*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^5)-6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)+a[-1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/(diff(G(xi), xi))^4-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/(diff(G(xi), xi))^3-a[1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

(19)

NULL

L := eval(ode, {K, K1, K2, K3})

c[1]^3*c[3]*(6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^4*(diff(G(xi), xi))^6)-12*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^5)-6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^3*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^4)+6*a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^3)+a[-1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^3/(diff(G(xi), xi))^4-6*a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)^2*(2*m*mu+lambda)/(diff(G(xi), xi))^3-a[1]*(2*m*mu+lambda)^2*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2)+3*c[2]^2*(a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2)-4*c[4]*c[1]*(a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2)+3*c[1]^2*c[3]*(a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2)^2 = 0

(20)

"collect(L,(m+1/(diff(G(xi),xi))))^( )"

Error, (in collect) cannot collect m+1/diff(G(xi),xi)

 
 

NULL

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