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Question: How solve this Special ODE ?

Question:How solve this Special ODE ?

salim-barzani 1785
ode differential-equations laplace adomian-decomposition
November 04 2024
0
This question is related to the Question How collect series solution of PDE?

I upload picture of solution and i try to solve but i fail i don't know how maple can do that just take laplace of one sidehow posible ?


 

> 

restart
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with(PDEtools)
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with(LinearAlgebra)
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NULL
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with(inttrans)
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with(SolveTools)
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undeclare(prime)

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

 (1)
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NULL
> 

eq := diff(y(x), `$`(x, 2))+(1+x)*(diff(y(x), x))-y(x) = 2*y(x)^3

diff(diff(y(x), x), x)+(1+x)*(diff(y(x), x))-y(x) = 2*y(x)^3

 (2)
> 

eq1 := laplace(eq, x, s)

s^2*laplace(y(x), x, s)-(D(y))(0)-s*y(0)-2*laplace(y(x), x, s)+s*laplace(y(x)*x, x, s)+s*laplace(y(x), x, s)-y(0) = 2*laplace(y(x)^3, x, s)

 (3)
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NULL
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NULL
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NULL
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NULL
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NULL
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NULL
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NULL
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NULL
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NULL
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restart
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with(inttrans)

[addtable, fourier, fouriercos, fouriersin, hankel, hilbert, invfourier, invhilbert, invlaplace, invmellin, laplace, mellin, savetable, setup]

 (4)
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eq := diff(y(x, t), x)+y(x, t)^2 = 1

diff(y(x, t), x)+y(x, t)^2 = 1

 (5)
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eq1 := laplace(eq, x, s)

s*laplace(y(x, t), x, s)-y(0, t)+laplace(y(x, t)^2, x, s) = 1/s

 (6)
> 

eq2 := subs(y(0, t) = 3, eq1)

s*laplace(y(x, t), x, s)-3+laplace(y(x, t)^2, x, s) = 1/s

 (7)
> 

lap := s*laplace(y(x, t), x, s) = 1/s+3-laplace(y(x, t)^2, x, s)

s*laplace(y(x, t), x, s) = 1/s+3-laplace(y(x, t)^2, x, s)

 (8)
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eq3 := lap/s

laplace(y(x, t), x, s) = (1/s+3-laplace(y(x, t)^2, x, s))/s

 (9)
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expand(%)

laplace(y(x, t), x, s) = 1/s^2+3/s-laplace(y(x, t)^2, x, s)/s

 (10)
> 

Geq := y(x, t) = invlaplace(1/s^2+3/s, s, x)-invlaplace(laplace(y(x, t)^2, x, s)/s, x, s)

y(x, t) = x+3-invlaplace(laplace(y(x, t)^2, x, s), x, s)/s

 (11)
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NULL
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k := K

K

 (12)
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f := proc (y) options operator, arrow; y^2 end proc

proc (y) options operator, arrow; y^2 end proc

 (13)
> 

for j from 0 to 4 do A[j] := subs(lambda = 0, (diff(f(sum(lambda^i*y[i](x), i = 0 .. 20)), [`$`(lambda, j)]))/factorial(j)) end do

y[0](x)^2

  

2*y[0](x)*y[1](x)

  

y[1](x)^2+2*y[0](x)*y[2](x)

  

2*y[1](x)*y[2](x)+2*y[0](x)*y[3](x)

  

y[2](x)^2+2*y[1](x)*y[3](x)+2*y[0](x)*y[4](x)

 (14)
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" y[0](x):=x+3"

proc (x) options operator, arrow, function_assign; 3+x end proc

 (15)
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lapy[1] := -laplace(A[0], x, s)/s

-(9*s^2+6*s+2)/s^4

 (16)
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simplify(%)

(-9*s^2-6*s-2)/s^4

 (17)
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invlaplace(%, s, x)

-(1/3)*x*(x^2+9*x+27)

 (18)
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simplify(-(1/3)*x*(x^2+9*x+27))

-(1/3)*x*(x^2+9*x+27)

 (19)
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normal(-(1/3)*x*(x^2+9*x+27), ':-expanded')

-(1/3)*x^3-3*x^2-9*x

 (20)
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"y[1](x):=-1/3 x^3-3 x^2-9 x"

proc (x) options operator, arrow, function_assign; -(1/3)*x^3-3*x^2-9*x end proc

 (21)
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lapy[2] := -laplace(A[1], x, s)/s

-(-72/s^3-48/s^4-16/s^5-54/s^2)/s

 (22)
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simplify(-(-72/s^3-48/s^4-16/s^5-54/s^2)/s)

(54*s^3+72*s^2+48*s+16)/s^6

 (23)
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expand(%)

54/s^3+72/s^4+48/s^5+16/s^6

 (24)
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invlaplace(%, s, x)

(1/15)*x^2*(2*x^3+30*x^2+180*x+405)

 (25)
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expand(%)

(2/15)*x^5+2*x^4+12*x^3+27*x^2

 (26)
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NULL
> 

NULL


 

Download laplace_of_special_ode.mw

example 1 i could't solve but example to i did
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