Internal problem ID [8663]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1083.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {y^{\prime } f^{\prime }\relax (x )}{f \relax (x )}+\left (\frac {3 f^{\prime }\relax (x )^{2}}{4 f \relax (x )^{2}}-\frac {f^{\prime \prime }\relax (x )}{2 f \relax (x )}-\frac {3 g^{\prime \prime }\relax (x )^{2}}{4 g^{\prime }\relax (x )^{2}}+\frac {g^{\prime \prime \prime }\relax (x )}{2 g^{\prime }\relax (x )}+\frac {\left (\frac {1}{4}-v^{2}\right ) g^{\prime }\relax (x )^{2}}{g \relax (x )^{2}}+g^{\prime }\relax (x )^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 43
dsolve(diff(diff(y(x),x),x)-diff(f(x),x)*diff(y(x),x)/f(x)+(3/4*diff(f(x),x)^2/f(x)^2-1/2*diff(diff(f(x),x),x)/f(x)-3/4*diff(diff(g(x),x),x)^2/diff(g(x),x)^2+1/2*diff(diff(diff(g(x),x),x),x)/diff(g(x),x)+(1/4-v^2)*diff(g(x),x)^2/g(x)^2+diff(g(x),x)^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {\frac {g \relax (x ) f \relax (x )}{\frac {d}{d x}g \relax (x )}}\, \BesselJ \left (v , g \relax (x )\right )+c_{2} \sqrt {\frac {g \relax (x ) f \relax (x )}{\frac {d}{d x}g \relax (x )}}\, \BesselY \left (v , g \relax (x )\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-((Derivative[1][f][x]*y'[x])/f[x]) + y[x]*((3*Derivative[1][f][x]^2)/(4*f[x]^2) + (g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/(2*f[x]) - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved