Internal problem ID [8007]
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]
\[ \boxed {a x {y^{\prime }}^{2}+\left (b x -y a +c \right ) y^{\prime }-b y=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 85
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 \left (x \right ) = \frac {-x b +c -2 \sqrt {-b c x}}{a} \\ y \left (x \right ) = \frac {-x b +c +2 \sqrt {-b c x}}{a} \\ y \left (x \right ) = -\frac {\left (a \,c_{1}^{2}+c_{1} b \right ) x}{-c_{1} a -b}-\frac {c c_{1}}{-c_{1} a -b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.017 (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*}