Internal problem ID [10008]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1677 (book 6.86).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2}=b \,x^{2}} \]
✓ Solution by Maple
Time used: 0.093 (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)
\[ y \left (x \right ) = \left (\int \left (-\frac {\sqrt {-b a}\, c_{1} \operatorname {BesselY}\left (1, \sqrt {-b a}\, x \right )}{x a \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-b a}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-b a}\, x \right )\right )}-\frac {\operatorname {BesselJ}\left (1, \sqrt {-b a}\, x \right ) \sqrt {-b a}}{x a \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-b a}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-b a}\, x \right )\right )}\right )d x +c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 120.409 (sec). Leaf size: 118
DSolve[-(b*x^2) + a*(-y[x] + x*y'[x])^2 + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (\int _1^x\frac {i \sqrt {b} \left (\operatorname {BesselY}\left (1,-i \sqrt {a} \sqrt {b} K[1]\right )-\operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right )}{\sqrt {a} \left (\operatorname {BesselY}\left (0,-i \sqrt {a} \sqrt {b} K[1]\right )+\operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right ) K[1]}dK[1]+c_2\right ) \]