Internal problem ID [9255]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1676.
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 }+a \left (y^{\prime } x -y\right )^{2}-b \,x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 110
dsolve(x^2*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2-b*x^2=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\int \left (-\frac {\sqrt {-a b}\, c_{1} \BesselY \left (1, \sqrt {-a b}\, x \right )}{x a \left (c_{1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}-\frac {\BesselJ \left (1, \sqrt {-a b}\, x \right ) \sqrt {-a b}}{x a \left (c_{1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}\right )d x +c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.232 (sec). Leaf size: 112
DSolve[-(b*x^2) + a*(-y[x] + x*y'[x])^2 + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (\int _1^x\frac {\frac {2 i \sqrt {b} Y_1\left (-i \sqrt {a} \sqrt {b} K[2]\right )}{\sqrt {a}}+b c_1 \, _0\tilde {F}_1\left (;2;\frac {1}{4} a b K[2]^2\right ) K[2]}{2 Y_0\left (-i \sqrt {a} \sqrt {b} K[2]\right ) K[2]+2 c_1 \, _0\tilde {F}_1\left (;1;\frac {1}{4} a b K[2]^2\right ) K[2]}dK[2]+c_2\right ) \\ \end{align*}