Internal problem ID [8609]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1029.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (f \relax (x )^{2}+f^{\prime }\relax (x )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 22
dsolve(diff(diff(y(x),x),x)-(f(x)^2+diff(f(x),x))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\int {\mathrm e}^{\int -2 f \relax (x )d x}d x +c_{1}\right ) {\mathrm e}^{\int f \relax (x )d x} c_{2} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 58
DSolve[-(y[x]*(f[x]^2 + Derivative[1][f][x])) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \exp \left (\int _1^xf(K[1])dK[1]\right )+c_2 \exp \left (\int _1^xf(K[2])dK[2]\right ) \int _1^x\exp \left (\int _1^{K[4]}-2 f(K[3])dK[3]\right )dK[4] \\ \end{align*}