Internal problem ID [8765]
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]
\[ \boxed {a x {y^{\prime }}^{2}-\left (a y+b x -a -b \right ) y^{\prime }+b y=0} \]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 96
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 \left (x \right ) = \frac {x b +a +b -2 \sqrt {a b x +b^{2} x}}{a} y \left (x \right ) = \frac {x b +a +b +2 \sqrt {a b x +b^{2} x}}{a} y \left (x \right ) = \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} \end{align*}
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 90
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 {-2 \sqrt {a^2 b x (a+b)}+a^2+a b (x+1)}{a^2} y(x)\to \frac {2 \sqrt {a^2 b x (a+b)}+a^2+a b (x+1)}{a^2} \end{align*}