Internal problem ID [8008]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 428.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
Solve \begin {gather*} \boxed {a x \left (y^{\prime }\right )^{2}+\left (x b -a y+c \right ) y^{\prime }-b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 62
dsolve(a*x*diff(y(x),x)^2+(b*x-a*y(x)+c)*diff(y(x),x)-b*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\left (a c_{1}^{2}+b c_{1}\right ) x}{-a c_{1}-b}-\frac {c c_{1}}{-a c_{1}-b} \\ y \relax (x ) = -\frac {b x}{a}+\frac {c}{a}+\sqrt {x}\, c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 80
DSolve[-(b*y[x]) + (c + b*x - a*y[x])*y'[x] + a*x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (x+\frac {c}{b+a c_1}\right ) \\ y(x)\to \frac {\left (\sqrt {c}-i \sqrt {b} \sqrt {x}\right )^2}{a} \\ y(x)\to \frac {\left (\sqrt {c}+i \sqrt {b} \sqrt {x}\right )^2}{a} \\ \end{align*}