Internal problem ID [8660]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1080.
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 (\frac {f^{\prime }\relax (x )}{f \relax (x )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\relax (x )}{f \relax (x )}+a^{2}-b^{2} f \relax (x )^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 74
dsolve(diff(diff(y(x),x),x)-(diff(f(x),x)/f(x)+2*a)*diff(y(x),x)+(a*diff(f(x),x)/f(x)+a^2-b^2*f(x)^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\int -\frac {{\mathrm e}^{\int -2 f \relax (x ) b d x} {\mathrm e}^{2 c_{1} b} f \relax (x ) b -{\mathrm e}^{\int -2 f \relax (x ) b d x} {\mathrm e}^{2 c_{1} b} a +f \relax (x ) b +a}{{\mathrm e}^{\int -2 f \relax (x ) b d x} {\mathrm e}^{2 c_{1} b}-1}d x} c_{2} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 47
DSolve[y[x]*(a^2 - b^2*f[x]^2 + (a*Derivative[1][f][x])/f[x]) - (2*a + Derivative[1][f][x]/f[x])*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{a x} \left (c_1 \exp \left (b \int _1^xf(K[1])dK[1]\right )+c_2 \exp \left (-b \int _1^xf(K[2])dK[2]\right )\right ) \\ \end{align*}