Internal problem ID [8659]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1079.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {a f^{\prime }\relax (x ) y^{\prime }}{f \relax (x )}+b f \relax (x )^{2 a} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x)-a*diff(f(x),x)/f(x)*diff(y(x),x)+b*f(x)^(2*a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\int i f \relax (x )^{a} \sqrt {b}d x}+c_{2} {\mathrm e}^{-\left (\int i f \relax (x )^{a} \sqrt {b}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 167
DSolve[b*f[x]^(2*a)*y[x] - (a*Derivative[1][f][x]*y'[x])/f[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {c_1} \sinh \left (\int _1^x-i \sqrt {b} f(K[1])^adK[1]+c_2\right ) \\ y(x)\to \sqrt {2} \sqrt {c_1} \sinh \left (\int _1^x-i \sqrt {b} f(K[1])^adK[1]+c_2\right ) \\ y(x)\to -\sqrt {2} \sqrt {c_1} \sinh \left (\int _1^xi \sqrt {b} f(K[2])^adK[2]+c_2\right ) \\ y(x)\to \sqrt {2} \sqrt {c_1} \sinh \left (\int _1^xi \sqrt {b} f(K[2])^adK[2]+c_2\right ) \\ \end{align*}