Internal problem ID [9253]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1674.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 27
dsolve(x^2*diff(diff(y(x),x),x)-(2*a+b-1)*x*diff(y(x),x)+(c^2*b^2*x^(2*b)+a*(a+b))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (x^{b} c \right ) x^{a}+c_{2} \cos \left (x^{b} c \right ) x^{a} \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 63
DSolve[(a*(a + b) + b^2*c^2*x^(2*b))*y[x] - (-1 + 2*a + b)*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{-\frac {a+b}{b}} c^{a/b} \left (x^{2 b}\right )^{\frac {a}{2 b}} \left (c_2 \sin \left (c \sqrt {x^{2 b}}\right )+2 c_1 \cos \left (c x^b\right )\right ) \\ \end{align*}