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Question: Help with the Solution Steps?

Question:Help with the Solution Steps?

delvin 55 Maple 2020
ode substitution homework differential-equations duplicate-question hpm
March 12 2025
0

Hi,

In order to obtain an algebraic system, one must set the coeffcients of (H + G′/G2)i to zero. Solve the obtained algebraic system.

But the expressions were not arranged correctly, but no answer was obtained, while the answer was as follows:

 

``NULL

> 

restart
> 

with(PDEtools):
df:= diff(diff(G(xi), xi)/(G(xi)^2), xi)= A+B*(diff(G(xi), xi)/(G(xi)^2))^2+ c*(diff(G(xi), xi)/(G(xi)^2));

(diff(diff(G(xi), xi), xi))/G(xi)^2-2*(diff(G(xi), xi))^2/G(xi)^3 = A+B*(diff(G(xi), xi))^2/G(xi)^4+c*(diff(G(xi), xi))/G(xi)^2

 (1)
> 

a := [a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10]:
> 

 

NULL

> 

p:= -2: q:= 2:
> 

Y1 :=xi -> (add(a[i+3]*(H+(diff(G(xi), xi)/(G(xi)^2)))^i, i = p .. q)):

NULL

> 

eq1 := -4*(k^2)*m*diff(Y1(xi), xi,xi) - 4*l*(Y1(xi)^2)+(4*(nu^2)-4*nu*n+n^2-4)*Y1(xi):
> 

eq2:=subs(df,eq1);

-4*k^2*m*(6*a0*(A+B*(diff(G(xi), xi))^2/G(xi)^4+c*(diff(G(xi), xi))/G(xi)^2)^2/(H+(diff(G(xi), xi))/G(xi)^2)^4-2*a0*((diff(diff(diff(G(xi), xi), xi), xi))/G(xi)^2-6*(diff(diff(G(xi), xi), xi))*(diff(G(xi), xi))/G(xi)^3+6*(diff(G(xi), xi))^3/G(xi)^4)/(H+(diff(G(xi), xi))/G(xi)^2)^3+2*a1*(A+B*(diff(G(xi), xi))^2/G(xi)^4+c*(diff(G(xi), xi))/G(xi)^2)^2/(H+(diff(G(xi), xi))/G(xi)^2)^3-a1*((diff(diff(diff(G(xi), xi), xi), xi))/G(xi)^2-6*(diff(diff(G(xi), xi), xi))*(diff(G(xi), xi))/G(xi)^3+6*(diff(G(xi), xi))^3/G(xi)^4)/(H+(diff(G(xi), xi))/G(xi)^2)^2+a3*((diff(diff(diff(G(xi), xi), xi), xi))/G(xi)^2-6*(diff(diff(G(xi), xi), xi))*(diff(G(xi), xi))/G(xi)^3+6*(diff(G(xi), xi))^3/G(xi)^4)+2*a4*(A+B*(diff(G(xi), xi))^2/G(xi)^4+c*(diff(G(xi), xi))/G(xi)^2)^2+2*a4*(H+(diff(G(xi), xi))/G(xi)^2)*((diff(diff(diff(G(xi), xi), xi), xi))/G(xi)^2-6*(diff(diff(G(xi), xi), xi))*(diff(G(xi), xi))/G(xi)^3+6*(diff(G(xi), xi))^3/G(xi)^4))-4*l*(a0/(H+(diff(G(xi), xi))/G(xi)^2)^2+a1/(H+(diff(G(xi), xi))/G(xi)^2)+a2+a3*(H+(diff(G(xi), xi))/G(xi)^2)+a4*(H+(diff(G(xi), xi))/G(xi)^2)^2)^2+(n^2-4*n*nu+4*nu^2-4)*(a0/(H+(diff(G(xi), xi))/G(xi)^2)^2+a1/(H+(diff(G(xi), xi))/G(xi)^2)+a2+a3*(H+(diff(G(xi), xi))/G(xi)^2)+a4*(H+(diff(G(xi), xi))/G(xi)^2)^2)

 (2)
> 

simplify(eq2):
> 

fin1:=simplify(numer(%)):

``

> 

for i from 0 to degree(fin1,H+(diff(G(xi), xi)/(G(xi)^2))) do EQ[i]:=simplify(coeff(fin1,H+(diff(G(xi), xi)/(G(xi)^2)),i)); end do;

4*k^2*(H*G(xi)^2+diff(G(xi), xi))*(-2*(diff(G(xi), xi))^4*a4-G(xi)^2*(8*H*a4+a3)*(diff(G(xi), xi))^3-3*G(xi)^4*H*(4*H*a4+a3)*(diff(G(xi), xi))^2+G(xi)^6*(-8*H^3*a4-3*H^2*a3+a1)*(diff(G(xi), xi))+G(xi)^8*(-2*H^4*a4-H^3*a3+H*a1+2*a0))*m*G(xi)^4*(diff(diff(diff(G(xi), xi), xi), xi))-24*(diff(G(xi), xi))*k^2*(H*G(xi)^2+diff(G(xi), xi))*(-2*(diff(G(xi), xi))^4*a4-G(xi)^2*(8*H*a4+a3)*(diff(G(xi), xi))^3-3*G(xi)^4*H*(4*H*a4+a3)*(diff(G(xi), xi))^2+G(xi)^6*(-8*H^3*a4-3*H^2*a3+a1)*(diff(G(xi), xi))+G(xi)^8*(-2*H^4*a4-H^3*a3+H*a1+2*a0))*m*G(xi)^3*(diff(diff(G(xi), xi), xi))-4*a4*(12*k^2*m*G(xi)^2+2*B^2*m*k^2+a4*l)*(diff(G(xi), xi))^8-8*G(xi)^2*(3*k^2*m*(10*H*a4+a3)*G(xi)^2+a4*((4*B^2*k^2*m+4*a4*l)*H+2*c*B*m*k^2+a3*l))*(diff(G(xi), xi))^7-16*(6*H*k^2*m*(5*H*a4+a3)*G(xi)^2+(3*B^2*a4*k^2*m+7*a4^2*l)*H^2+(7/2)*((8/7)*c*B*m*k^2+a3*l)*a4*H+(m*(B*A+(1/2)*c^2)*k^2+(1/2)*l*a2-(1/4)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(1/4)*a3^2*l)*G(xi)^4*(diff(G(xi), xi))^6-64*G(xi)^6*(-(3/8)*k^2*m*(-20*H^3*a4-6*H^2*a3+a1)*G(xi)^2+((1/2)*k^2*m*B^2*a4+(7/2)*a4^2*l)*H^3+(21/8)*((4/7)*c*B*m*k^2+a3*l)*a4*H^2+((m*(B*A+(1/2)*c^2)*k^2+(3/4)*l*a2-(3/8)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(3/8)*a3^2*l)*H+((1/4)*c*A*m*k^2+(1/8)*a1*l)*a4+(1/8)*a1*B^2*k^2*m+(1/8)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*(diff(G(xi), xi))^5-8*G(xi)^8*(-6*k^2*m*(-5*H^4*a4-2*H^3*a3+H*a1+a0)*G(xi)^2+(B^2*a4*k^2*m+35*a4^2*l)*H^4+35*((8/35)*c*B*m*k^2+a3*l)*a4*H^3+(((12*A*B+6*c^2)*m*k^2+15*l*a2-(15/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(15/2)*a3^2*l)*H^2+((8*A*c*k^2*m+5*a1*l)*a4+a1*B^2*k^2*m+5*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H+(A^2*k^2*m+a0*l)*a4+(3*B^2*a0+2*B*a1*c)*m*k^2+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*(diff(G(xi), xi))^4-32*G(xi)^10*(-(3/4)*H*k^2*m*(-2*H^4*a4-H^3*a3+H*a1+2*a0)*G(xi)^2+7*H^5*a4^2*l+(35/4)*((2/35)*c*B*m*k^2+a3*l)*a4*H^4+((m*(2*A*B+c^2)*k^2+5*l*a2-(5/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(5/2)*a3^2*l)*H^3+(((5/2)*a1*l+3*c*A*m*k^2)*a4+(5/2)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^2+((A^2*k^2*m+a0*l)*a4+(1/2)*k^2*m*B*a1*c+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H+(1/2)*((B*A+(1/2)*c^2)*a1+3*B*a0*c)*m*k^2+((1/4)*a0*a3+(1/4)*a1*a2)*l-(1/8)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*(diff(G(xi), xi))^3-48*((7/3)*H^6*a4^2*l+(7/2)*H^5*a3*a4*l+(((1/3)*m*(B*A+(1/2)*c^2)*k^2+(5/2)*l*a2-(5/4)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(5/4)*a3^2*l)*H^4+(((5/3)*a1*l+(4/3)*c*A*m*k^2)*a4+(5/3)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^3+((A^2*k^2*m+a0*l)*a4+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^2+((1/3)*(B*A+(1/2)*c^2)*a1*m*k^2+((1/2)*a0*a3+(1/2)*a1*a2)*l-(1/4)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H+(1/3)*m*(A*a1*c+3*(B*A+(1/2)*c^2)*a0)*k^2+((1/6)*a0*a2+(1/12)*a1^2)*l-(1/12)*(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*G(xi)^12*(diff(G(xi), xi))^2-8*(4*a4^2*H^7*l+7*a3*a4*H^6*l+((6*l*a2+3*nu*n-3*nu^2-(3/4)*n^2+3)*a4+3*a3^2*l)*H^5+((2*A*c*k^2*m+5*a1*l)*a4+5*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^4+((4*A^2*k^2*m+4*a0*l)*a4+(4*a1*a3+2*a2^2)*l-2*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^3+((3*a0*a3+3*a1*a2)*l-(3/2)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H^2+(2*k^2*m*A*a1*c+(2*a0*a2+a1^2)*l-(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*H+A*m*(A*a1+6*a0*c)*k^2+a1*l*a0)*G(xi)^14*(diff(G(xi), xi))-8*G(xi)^16*((1/2)*H^8*a4^2*l+H^7*a3*a4*l+((l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(1/2)*a3^2*l)*H^6+(a1*a4*l+a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^5+((A^2*k^2*m+a0*l)*a4+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^4+((a0*a3+a1*a2)*l-(1/2)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H^3+(((1/2)*a1^2+a0*a2)*l-(1/2)*(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*H^2+a1*(A^2*k^2*m+a0*l)*H+(1/2)*a0^2*l+3*k^2*m*A^2*a0)

 (3)
> 

 

NULL

> 

for i from 0 to degree(fin1,H+(diff(G(xi), xi)/(G(xi)^2))) do EQ[i]:=simplify(coeff(fin1,H+(diff(G(xi), xi)/(G(xi)^2)),i)); end do;

4*k^2*(H*G(xi)^2+diff(G(xi), xi))*(-2*(diff(G(xi), xi))^4*a4-G(xi)^2*(8*H*a4+a3)*(diff(G(xi), xi))^3-3*G(xi)^4*H*(4*H*a4+a3)*(diff(G(xi), xi))^2+G(xi)^6*(-8*H^3*a4-3*H^2*a3+a1)*(diff(G(xi), xi))+G(xi)^8*(-2*H^4*a4-H^3*a3+H*a1+2*a0))*m*G(xi)^4*(diff(diff(diff(G(xi), xi), xi), xi))-24*(diff(G(xi), xi))*k^2*(H*G(xi)^2+diff(G(xi), xi))*(-2*(diff(G(xi), xi))^4*a4-G(xi)^2*(8*H*a4+a3)*(diff(G(xi), xi))^3-3*G(xi)^4*H*(4*H*a4+a3)*(diff(G(xi), xi))^2+G(xi)^6*(-8*H^3*a4-3*H^2*a3+a1)*(diff(G(xi), xi))+G(xi)^8*(-2*H^4*a4-H^3*a3+H*a1+2*a0))*m*G(xi)^3*(diff(diff(G(xi), xi), xi))-4*a4*(12*k^2*m*G(xi)^2+2*B^2*m*k^2+a4*l)*(diff(G(xi), xi))^8-8*G(xi)^2*(3*k^2*m*(10*H*a4+a3)*G(xi)^2+a4*((4*B^2*k^2*m+4*a4*l)*H+2*c*B*m*k^2+a3*l))*(diff(G(xi), xi))^7-16*(6*H*k^2*m*(5*H*a4+a3)*G(xi)^2+(3*B^2*a4*k^2*m+7*a4^2*l)*H^2+(7/2)*((8/7)*c*B*m*k^2+a3*l)*a4*H+(m*(B*A+(1/2)*c^2)*k^2+(1/2)*l*a2-(1/4)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(1/4)*a3^2*l)*G(xi)^4*(diff(G(xi), xi))^6-64*G(xi)^6*(-(3/8)*k^2*m*(-20*H^3*a4-6*H^2*a3+a1)*G(xi)^2+((1/2)*k^2*m*B^2*a4+(7/2)*a4^2*l)*H^3+(21/8)*((4/7)*c*B*m*k^2+a3*l)*a4*H^2+((m*(B*A+(1/2)*c^2)*k^2+(3/4)*l*a2-(3/8)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(3/8)*a3^2*l)*H+((1/4)*c*A*m*k^2+(1/8)*a1*l)*a4+(1/8)*a1*B^2*k^2*m+(1/8)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*(diff(G(xi), xi))^5-8*G(xi)^8*(-6*k^2*m*(-5*H^4*a4-2*H^3*a3+H*a1+a0)*G(xi)^2+(B^2*a4*k^2*m+35*a4^2*l)*H^4+35*((8/35)*c*B*m*k^2+a3*l)*a4*H^3+(((12*A*B+6*c^2)*m*k^2+15*l*a2-(15/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(15/2)*a3^2*l)*H^2+((8*A*c*k^2*m+5*a1*l)*a4+a1*B^2*k^2*m+5*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H+(A^2*k^2*m+a0*l)*a4+(3*B^2*a0+2*B*a1*c)*m*k^2+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*(diff(G(xi), xi))^4-32*G(xi)^10*(-(3/4)*H*k^2*m*(-2*H^4*a4-H^3*a3+H*a1+2*a0)*G(xi)^2+7*H^5*a4^2*l+(35/4)*((2/35)*c*B*m*k^2+a3*l)*a4*H^4+((m*(2*A*B+c^2)*k^2+5*l*a2-(5/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(5/2)*a3^2*l)*H^3+(((5/2)*a1*l+3*c*A*m*k^2)*a4+(5/2)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^2+((A^2*k^2*m+a0*l)*a4+(1/2)*k^2*m*B*a1*c+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H+(1/2)*((B*A+(1/2)*c^2)*a1+3*B*a0*c)*m*k^2+((1/4)*a0*a3+(1/4)*a1*a2)*l-(1/8)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*(diff(G(xi), xi))^3-48*((7/3)*H^6*a4^2*l+(7/2)*H^5*a3*a4*l+(((1/3)*m*(B*A+(1/2)*c^2)*k^2+(5/2)*l*a2-(5/4)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(5/4)*a3^2*l)*H^4+(((5/3)*a1*l+(4/3)*c*A*m*k^2)*a4+(5/3)*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^3+((A^2*k^2*m+a0*l)*a4+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^2+((1/3)*(B*A+(1/2)*c^2)*a1*m*k^2+((1/2)*a0*a3+(1/2)*a1*a2)*l-(1/4)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H+(1/3)*m*(A*a1*c+3*(B*A+(1/2)*c^2)*a0)*k^2+((1/6)*a0*a2+(1/12)*a1^2)*l-(1/12)*(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*G(xi)^12*(diff(G(xi), xi))^2-8*(4*a4^2*H^7*l+7*a3*a4*H^6*l+((6*l*a2+3*nu*n-3*nu^2-(3/4)*n^2+3)*a4+3*a3^2*l)*H^5+((2*A*c*k^2*m+5*a1*l)*a4+5*a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^4+((4*A^2*k^2*m+4*a0*l)*a4+(4*a1*a3+2*a2^2)*l-2*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^3+((3*a0*a3+3*a1*a2)*l-(3/2)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H^2+(2*k^2*m*A*a1*c+(2*a0*a2+a1^2)*l-(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*H+A*m*(A*a1+6*a0*c)*k^2+a1*l*a0)*G(xi)^14*(diff(G(xi), xi))-8*G(xi)^16*((1/2)*H^8*a4^2*l+H^7*a3*a4*l+((l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1))*a4+(1/2)*a3^2*l)*H^6+(a1*a4*l+a3*(l*a2-(1/2)*(nu-(1/2)*n-1)*(nu-(1/2)*n+1)))*H^5+((A^2*k^2*m+a0*l)*a4+(a1*a3+(1/2)*a2^2)*l-(1/2)*(nu-(1/2)*n-1)*a2*(nu-(1/2)*n+1))*H^4+((a0*a3+a1*a2)*l-(1/2)*(nu-(1/2)*n-1)*a1*(nu-(1/2)*n+1))*H^3+(((1/2)*a1^2+a0*a2)*l-(1/2)*(nu-(1/2)*n-1)*a0*(nu-(1/2)*n+1))*H^2+a1*(A^2*k^2*m+a0*l)*H+(1/2)*a0^2*l+3*k^2*m*A^2*a0)

 (4)
> 

Eqs:={seq(EQ[i],i=0..12)}:

 

> 

sol:=solve(Eqs,{a0, a1, a2, a3, a4, H, nu},explicit)

(5)
 

 

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