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Question: How can we generate solutions for an ODE using constants?

Question:How can we generate solutions for an ODE using constants?

salim-barzani 1785 Maple 2024
ode parameters dsolve spam differential-equations duplicate-question
October 26 2024
0

In many papers, I've noticed that the solution to an ODE (ordinary differential equation) often emerges directly when there's only a single function involved. My question is: is there a way to generate solutions to an ODE by producing specific parameters?

 

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

with(Physics)
> 

with(SolveTools)
> 

undeclare(prime)

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

 (1)
> 

NULL
> 

S := diff(F(xi), xi) = sqrt(P*F(xi)^4+Q*F(xi)^2+R)

diff(F(xi), xi) = (P*F(xi)^4+Q*F(xi)^2+R)^(1/2)

 (2)
> 

S1 := dsolve(S, F(xi))

xi-Intat(1/(P*_a^4+Q*_a^2+R)^(1/2), _a = F(xi))+c__1 = 0

 (3)
> 

S2 := (diff(F(xi), xi))^2 = P*F(xi)^4+Q*F(xi)^2+R

(diff(F(xi), xi))^2 = P*F(xi)^4+Q*F(xi)^2+R

 (4)
> 

S3 := dsolve(S2, F(xi))

F(xi) = -(1/2)*(-2*P*(Q+(-4*P*R+Q^2)^(1/2)))^(1/2)/P, F(xi) = (1/2)*(-2*P*(Q+(-4*P*R+Q^2)^(1/2)))^(1/2)/P, F(xi) = -(1/2)*2^(1/2)*(P*(-Q+(-4*P*R+Q^2)^(1/2)))^(1/2)/P, F(xi) = (1/2)*2^(1/2)*(P*(-Q+(-4*P*R+Q^2)^(1/2)))^(1/2)/P, F(xi) = JacobiSN((1/2)*(2*(-4*P*R+Q^2)^(1/2)-2*Q)^(1/2)*xi+c__1, (-2*(Q*(-4*P*R+Q^2)^(1/2)+2*R*P-Q^2)*R*P)^(1/2)/(Q*(-4*P*R+Q^2)^(1/2)+2*R*P-Q^2))*R*2^(1/2)/(R*(-Q+(-4*P*R+Q^2)^(1/2)))^(1/2)

 (5)
 

NULL

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