Internal problem ID [9249]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1670.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x +a \left (y^{\prime } x -y\right )^{2}-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 35
dsolve(x*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2-b=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\int \frac {i \tan \left (-i \sqrt {a}\, \sqrt {b}\, x +c_{1}\right ) \sqrt {b}}{\sqrt {a}\, x^{2}}d x +c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 1.468 (sec). Leaf size: 50
DSolve[-b + a*(-y[x] + x*y'[x])^2 + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (\int _1^x\frac {\sqrt {-\frac {b}{a}} \tan \left (c_1+\frac {b K[2]}{\sqrt {-\frac {b}{a}}}\right )}{K[2]^2}dK[2]+c_2\right ) \\ \end{align*}