Internal problem ID [7964]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 384.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+\left (a x +b \right ) y^{\prime }-a y+c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.22 (sec). Leaf size: 50
dsolve(diff(y(x),x)^2+(a*x+b)*diff(y(x),x)-a*y(x)+c = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {-a^{2} x^{2}-2 b x a -b^{2}+4 c}{4 a} \\ y \relax (x ) = c_{1} x +\frac {c_{1} b +c_{1}^{2}+c}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 43
DSolve[c - a*y[x] + (b + a*x)*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c+c_1 (a x+b+c_1)}{a} \\ y(x)\to -\frac {(a x+b)^2-4 c}{4 a} \\ \end{align*}