Internal problem ID [3723]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 999.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {a^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) \left (y^{\prime }\right )^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.703 (sec). Leaf size: 195
dsolve(a^2*(b^2-(c*x-a*y(x))^2)*diff(y(x),x)^2+2*a*b^2*c*diff(y(x),x)+c^2*(b^2-(c*x-a*y(x))^2) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c x -\sqrt {2}\, b}{a} \\ y \relax (x ) = \frac {c x +\sqrt {2}\, b}{a} \\ y \relax (x ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {a \left (\textit {\_a}^{2} a^{2}-2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right ) c}d \textit {\_a} +c_{1}\right ) a +c x}{a} \\ y \relax (x ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {a \left (-\textit {\_a}^{2} a^{2}+2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right ) c}d \textit {\_a} +c_{1}\right ) a +c x}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.156 (sec). Leaf size: 71
DSolve[a^2 ( b^2 -(c x-a y[x])^2 ) (y'[x])^2 +2 a b^2 c y'[x]+c^2(b^2-(c x-a y[x])^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c c_1-\sqrt {b^2-c^2 (x-c_1){}^2}}{a} \\ y(x)\to \frac {\sqrt {b^2-c^2 (x-c_1){}^2}+c c_1}{a} \\ \end{align*}