Internal problem ID [3670]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 944.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}+2 y^{\prime } a x -a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 118
dsolve(y(x)*diff(y(x),x)^2+2*a*x*diff(y(x),x)-a*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = x \sqrt {-a} \\ y \relax (x ) = -x \sqrt {-a} \\ y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{2}+\sqrt {a \,\textit {\_a}^{2}+a^{2}}-a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {a \,\textit {\_a}^{2}+a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.667 (sec). Leaf size: 137
DSolve[y[x] (y'[x])^2+2 a x y'[x]-a y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to \sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to -\sqrt {e^{c_1} \left (2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to \sqrt {e^{c_1} \left (2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to -i \sqrt {a} x \\ y(x)\to i \sqrt {a} x \\ \end{align*}