Internal problem ID [8105]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 525.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-a x y y^{\prime }+2 y^{2} a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.259 (sec). Leaf size: 129
dsolve(diff(y(x),x)^2-a*x*y(x)*diff(y(x),x)+2*a*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-8 a}\right )^{-\frac {2 a}{\sqrt {a^{2}}}} {\mathrm e}^{\frac {x \sqrt {a^{2} x^{2}-8 a}}{4}+\frac {a \,x^{2}}{4}} \\ y \relax (x ) = c_{1} \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-8 a}\right )^{\frac {2 a}{\sqrt {a^{2}}}} {\mathrm e}^{-\frac {x \sqrt {a^{2} x^{2}-8 a}}{4}+\frac {a \,x^{2}}{4}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.27 (sec). Leaf size: 125
DSolve[2*a*y[x]^2 - a*x*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 e^{\frac {1}{4} \left (a x^2-\sqrt {a} x \sqrt {a x^2-8}\right )}}{\left (\sqrt {a x^2-8}-\sqrt {a} x\right )^2} \\ y(x)\to c_1 e^{\frac {1}{4} \left (a x^2+\sqrt {a} x \sqrt {a x^2-8}\right )} \left (\sqrt {a x^2-8}-\sqrt {a} x\right )^2 \\ y(x)\to 0 \\ \end{align*}