Internal problem ID [8658]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1078.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+f \relax (x ) y^{\prime }+\left (\frac {f \relax (x )^{2}}{4}+\frac {f^{\prime }\relax (x )}{2}+a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x)+f(x)*diff(y(x),x)+(1/4*f(x)^2+1/2*diff(f(x),x)+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {\left (\int f \relax (x )d x \right )}{2}} \sinh \left (\sqrt {-a}\, x \right )+c_{2} {\mathrm e}^{-\frac {\left (\int f \relax (x )d x \right )}{2}} \cosh \left (\sqrt {-a}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 69
DSolve[y[x]*(a + f[x]^2/4 + Derivative[1][f][x]/2) + f[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (2 \sqrt {a} c_1-i c_2 e^{2 i \sqrt {a} x}\right ) \exp \left (-\frac {1}{2} \int _1^xf(K[1])dK[1]-i \sqrt {a} x\right )}{2 \sqrt {a}} \\ \end{align*}