Internal problem ID [8009]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 429.
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 (a y+b x -a -b \right ) y^{\prime }+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.283 (sec). Leaf size: 63
dsolve(a*x*diff(y(x),x)^2-(a*y(x)+b*x-a-b)*diff(y(x),x)+b*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-a c_{1}^{2}+c_{1} b \right ) x}{-c_{1} a +b}+\frac {\left (-a -b \right ) c_{1}}{-c_{1} a +b} \\ y \relax (x ) = \frac {b x}{a}+c_{1} \sqrt {x}+1+\frac {b}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 88
DSolve[b*y[x] - (-a - b + 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 {a+b}{-b+a c_1}\right ) \\ y(x)\to \frac {a (a+b x+b)-2 \sqrt {a^2 b x (a+b)}}{a^2} \\ y(x)\to \frac {2 \sqrt {a^2 b x (a+b)}+a (a+b x+b)}{a^2} \\ \end{align*}