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How get two different solution of second order ODE for one parameter?

December 23, 2024, 9:42 am
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I faced this problem before but i didn't remember how solve this issue  for that parameter when i substitute and get solution How get two different solution  also V(xi) must satisfy the odetest

> 

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)
> 

declare(Omega(x, t)); declare(U(xi)); declare(V(xi)); declare(G(xi))

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

 

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

 

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

 

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

(2)
> 

ode := 4*V(xi)^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*V(xi)^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*V(xi)^2-2*V(xi)*(diff(diff(V(xi), xi), xi))*alpha*m*n+(-alpha*m^2+2*alpha*m*n)*(diff(V(xi), xi))^2 = 0

4*V(xi)^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*V(xi)^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*V(xi)^2-2*V(xi)*(diff(diff(V(xi), xi), xi))*alpha*m*n+(-alpha*m^2+2*alpha*m*n)*(diff(V(xi), xi))^2 = 0

(3)
> 

S := (diff(G(xi), xi))^2 = a*G(xi)^2+b*G(xi)^3+c*G(xi)^4

(diff(G(xi), xi))^2 = a*G(xi)^2+b*G(xi)^3+c*G(xi)^4

(4)
> 

S1 := sum(A[i]*G(xi)^i, i = 0 .. 1)

A[0]+A[1]*G(xi)

(5)
> 

S12 := diff(S1, xi)

A[1]*(diff(G(xi), xi))

(6)
> 

S123 := diff(G(xi), xi) = sqrt(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)

diff(G(xi), xi) = (a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(7)
> 

subs(S123, S12)

A[1]*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(8)
> 

S11 := %

A[1]*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(9)
> 

S2 := diff(S11, xi)

(1/2)*A[1]*(2*a*G(xi)*(diff(G(xi), xi))+3*b*G(xi)^2*(diff(G(xi), xi))+4*c*G(xi)^3*(diff(G(xi), xi)))/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(10)
> 

S22 := subs(S123, S2)

(1/2)*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(11)
> 

K := V(xi) = S1

V(xi) = A[0]+A[1]*G(xi)

(12)
> 

K1 := diff(V(xi), xi) = S11

diff(V(xi), xi) = A[1]*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(13)
> 

K2 := diff(V(xi), xi, xi) = S22

diff(diff(V(xi), xi), xi) = (1/2)*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(14)
> 

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

4*(A[0]+A[1]*G(xi))^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*(A[0]+A[1]*G(xi))^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*(A[0]+A[1]*G(xi))^2-(A[0]+A[1]*G(xi))*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))*alpha*m*n/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+(-alpha*m^2+2*alpha*m*n)*A[1]^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4) = 0

(15)
> 

numer(lhs(4*(A[0]+A[1]*G(xi))^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*(A[0]+A[1]*G(xi))^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*(A[0]+A[1]*G(xi))^2-(A[0]+A[1]*G(xi))*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))*alpha*m*n/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+(-alpha*m^2+2*alpha*m*n)*A[1]^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4) = 0))*denom(rhs(4*(A[0]+A[1]*G(xi))^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*(A[0]+A[1]*G(xi))^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*(A[0]+A[1]*G(xi))^2-(A[0]+A[1]*G(xi))*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))*alpha*m*n/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+(-alpha*m^2+2*alpha*m*n)*A[1]^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4) = 0)) = numer(rhs(4*(A[0]+A[1]*G(xi))^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*(A[0]+A[1]*G(xi))^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*(A[0]+A[1]*G(xi))^2-(A[0]+A[1]*G(xi))*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))*alpha*m*n/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+(-alpha*m^2+2*alpha*m*n)*A[1]^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4) = 0))*denom(lhs(4*(A[0]+A[1]*G(xi))^4*n^2*sigma+(-4*beta*k*m*n^2+4*gamma*k*m*n^2)*(A[0]+A[1]*G(xi))^3+(4*alpha*k^2*m^2*n^2-4*delta^2*m*n^2+4*m*n^2*w)*(A[0]+A[1]*G(xi))^2-(A[0]+A[1]*G(xi))*A[1]*(2*a*G(xi)*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+3*b*G(xi)^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+4*c*G(xi)^3*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2))*alpha*m*n/(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)+(-alpha*m^2+2*alpha*m*n)*A[1]^2*(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4) = 0))

-(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)*(-12*G(xi)^2*gamma*k*m*n^2*A[0]*A[1]^2+12*G(xi)^2*beta*k*m*n^2*A[0]*A[1]^2-8*G(xi)*alpha*k^2*m^2*n^2*A[0]*A[1]+4*G(xi)^3*alpha*c*m*n*A[0]*A[1]-12*G(xi)*gamma*k*m*n^2*A[0]^2*A[1]+12*G(xi)*beta*k*m*n^2*A[0]^2*A[1]+3*G(xi)^2*alpha*b*m*n*A[0]*A[1]+2*G(xi)*a*alpha*m*n*A[0]*A[1]-4*G(xi)^3*gamma*k*m*n^2*A[1]^3+4*G(xi)^3*beta*k*m*n^2*A[1]^3-4*G(xi)^2*alpha*k^2*m^2*n^2*A[1]^2+2*G(xi)^4*alpha*c*m*n*A[1]^2+G(xi)^3*alpha*b*m*n*A[1]^2+8*G(xi)*delta^2*m*n^2*A[0]*A[1]-8*G(xi)*m*n^2*w*A[0]*A[1]-4*n^2*sigma*A[0]^4+G(xi)^4*alpha*c*m^2*A[1]^2+4*beta*k*m*n^2*A[0]^3-4*alpha*k^2*m^2*n^2*A[0]^2+4*delta^2*m*n^2*A[0]^2-4*m*n^2*w*A[0]^2-4*G(xi)^4*n^2*sigma*A[1]^4+G(xi)^3*alpha*b*m^2*A[1]^2+4*G(xi)^2*delta^2*m*n^2*A[1]^2-24*G(xi)^2*n^2*sigma*A[0]^2*A[1]^2+G(xi)^2*a*alpha*m^2*A[1]^2-4*G(xi)^2*m*n^2*w*A[1]^2-16*G(xi)*n^2*sigma*A[0]^3*A[1]-4*gamma*k*m*n^2*A[0]^3-16*G(xi)^3*n^2*sigma*A[0]*A[1]^3) = 0

(16)
> 

%/(-sqrt(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4))

-12*G(xi)^2*gamma*k*m*n^2*A[0]*A[1]^2+12*G(xi)^2*beta*k*m*n^2*A[0]*A[1]^2-8*G(xi)*alpha*k^2*m^2*n^2*A[0]*A[1]+4*G(xi)^3*alpha*c*m*n*A[0]*A[1]-12*G(xi)*gamma*k*m*n^2*A[0]^2*A[1]+12*G(xi)*beta*k*m*n^2*A[0]^2*A[1]+3*G(xi)^2*alpha*b*m*n*A[0]*A[1]+2*G(xi)*a*alpha*m*n*A[0]*A[1]-4*G(xi)^3*gamma*k*m*n^2*A[1]^3+4*G(xi)^3*beta*k*m*n^2*A[1]^3-4*G(xi)^2*alpha*k^2*m^2*n^2*A[1]^2+2*G(xi)^4*alpha*c*m*n*A[1]^2+G(xi)^3*alpha*b*m*n*A[1]^2+8*G(xi)*delta^2*m*n^2*A[0]*A[1]-8*G(xi)*m*n^2*w*A[0]*A[1]-4*n^2*sigma*A[0]^4+G(xi)^4*alpha*c*m^2*A[1]^2+4*beta*k*m*n^2*A[0]^3-4*alpha*k^2*m^2*n^2*A[0]^2+4*delta^2*m*n^2*A[0]^2-4*m*n^2*w*A[0]^2-4*G(xi)^4*n^2*sigma*A[1]^4+G(xi)^3*alpha*b*m^2*A[1]^2+4*G(xi)^2*delta^2*m*n^2*A[1]^2-24*G(xi)^2*n^2*sigma*A[0]^2*A[1]^2+G(xi)^2*a*alpha*m^2*A[1]^2-4*G(xi)^2*m*n^2*w*A[1]^2-16*G(xi)*n^2*sigma*A[0]^3*A[1]-4*gamma*k*m*n^2*A[0]^3-16*G(xi)^3*n^2*sigma*A[0]*A[1]^3 = 0

(17)
> 

collect(%, G)

(-4*n^2*sigma*A[1]^4+alpha*c*m^2*A[1]^2+2*alpha*c*m*n*A[1]^2)*G(xi)^4+(4*beta*k*m*n^2*A[1]^3-4*gamma*k*m*n^2*A[1]^3-16*n^2*sigma*A[0]*A[1]^3+alpha*b*m^2*A[1]^2+alpha*b*m*n*A[1]^2+4*alpha*c*m*n*A[0]*A[1])*G(xi)^3+(-4*alpha*k^2*m^2*n^2*A[1]^2+12*beta*k*m*n^2*A[0]*A[1]^2-12*gamma*k*m*n^2*A[0]*A[1]^2+4*delta^2*m*n^2*A[1]^2-24*n^2*sigma*A[0]^2*A[1]^2+a*alpha*m^2*A[1]^2+3*alpha*b*m*n*A[0]*A[1]-4*m*n^2*w*A[1]^2)*G(xi)^2+(-8*alpha*k^2*m^2*n^2*A[0]*A[1]+12*beta*k*m*n^2*A[0]^2*A[1]-12*gamma*k*m*n^2*A[0]^2*A[1]+8*delta^2*m*n^2*A[0]*A[1]-16*n^2*sigma*A[0]^3*A[1]+2*a*alpha*m*n*A[0]*A[1]-8*m*n^2*w*A[0]*A[1])*G(xi)-4*alpha*k^2*m^2*n^2*A[0]^2+4*beta*k*m*n^2*A[0]^3-4*gamma*k*m*n^2*A[0]^3+4*delta^2*m*n^2*A[0]^2-4*n^2*sigma*A[0]^4-4*m*n^2*w*A[0]^2 = 0

(18)
> 

eq0 := -4*alpha*k^2*m^2*n^2*A[0]^2+4*beta*k*m*n^2*A[0]^3-4*gamma*k*m*n^2*A[0]^3+4*delta^2*m*n^2*A[0]^2-4*n^2*sigma*A[0]^4-4*m*n^2*w*A[0]^2 = 0

> 

eq1 := -8*alpha*k^2*m^2*n^2*A[0]*A[1]+12*beta*k*m*n^2*A[0]^2*A[1]-12*gamma*k*m*n^2*A[0]^2*A[1]+8*delta^2*m*n^2*A[0]*A[1]-16*n^2*sigma*A[0]^3*A[1]+2*a*alpha*m*n*A[0]*A[1]-8*m*n^2*w*A[0]*A[1] = 0

> 

eq2 := -4*alpha*k^2*m^2*n^2*A[1]^2+12*beta*k*m*n^2*A[0]*A[1]^2-12*gamma*k*m*n^2*A[0]*A[1]^2+4*delta^2*m*n^2*A[1]^2-24*n^2*sigma*A[0]^2*A[1]^2+a*alpha*m^2*A[1]^2+3*alpha*b*m*n*A[0]*A[1]-4*m*n^2*w*A[1]^2 = 0

> 

eq3 := 4*beta*k*m*n^2*A[1]^3-4*gamma*k*m*n^2*A[1]^3-16*n^2*sigma*A[0]*A[1]^3+alpha*b*m^2*A[1]^2+alpha*b*m*n*A[1]^2+4*alpha*c*m*n*A[0]*A[1] = 0

> 

eq4 := -4*n^2*sigma*A[1]^4+alpha*c*m^2*A[1]^2+2*alpha*c*m*n*A[1]^2 = 0

> 

C := solve({eq0, eq1, eq2, eq3, eq4}, {a, b, c, A[0]})

{a = 4*n^2*(alpha*k^2*m-delta^2+w)/(m*alpha), b = 4*(gamma-beta)*k*n^2*A[1]/((m+n)*alpha), c = 4*n^2*sigma*A[1]^2/(alpha*m*(m+2*n)), A[0] = 0}

(19)
> 

W := q(x, t) = V(xi)^(1/(2*n))

q(x, t) = V(xi)^((1/2)/n)

(20)
> 

S123 := diff(G(xi), xi) = sqrt(a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)

diff(G(xi), xi) = (a*G(xi)^2+b*G(xi)^3+c*G(xi)^4)^(1/2)

(21)
> 

dsolve(S, G(xi))

G(xi) = (1/2)*(-b+(-4*a*c+b^2)^(1/2))/c, G(xi) = -(1/2)*(b+(-4*a*c+b^2)^(1/2))/c, G(xi) = -4*a*exp(c__1*a^(1/2))/(exp(xi*a^(1/2))*(4*a*c-b^2+2*exp(c__1*a^(1/2))*b/exp(xi*a^(1/2))-(exp(c__1*a^(1/2)))^2/(exp(xi*a^(1/2)))^2)), G(xi) = -4*a*exp(xi*a^(1/2))/(exp(c__1*a^(1/2))*(4*a*c-b^2+2*exp(xi*a^(1/2))*b/exp(c__1*a^(1/2))-(exp(xi*a^(1/2)))^2/(exp(c__1*a^(1/2)))^2))

(22)
 

``

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