Internal problem ID [8082]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 502.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {\left (a y-b x \right )^{2} \left (\left (y^{\prime }\right )^{2} a^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.744 (sec). Leaf size: 195
dsolve((a*y(x)-b*x)^2*(a^2*diff(y(x),x)^2+b^2)-c^2*(a*diff(y(x),x)+b)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {b x -\sqrt {2}\, c}{a} \\ y \relax (x ) = \frac {b x +\sqrt {2}\, c}{a} \\ y \relax (x ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {a \left (\textit {\_a}^{2} a^{2}-2 c^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right ) b}d \textit {\_a} +c_{1}\right ) a +b x}{a} \\ y \relax (x ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {a \left (-\textit {\_a}^{2} a^{2}+2 c^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right ) b}d \textit {\_a} +c_{1}\right ) a +b x}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.792 (sec). Leaf size: 71
DSolve[-(c^2*(b + a*y'[x])^2) + (-(b*x) + a*y[x])^2*(b^2 + a^2*y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b c_1-\sqrt {c^2-b^2 (x-c_1){}^2}}{a} \\ y(x)\to \frac {\sqrt {c^2-b^2 (x-c_1){}^2}+b c_1}{a} \\ \end{align*}