Internal problem ID [8860]
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]
\[ \boxed {{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 98
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 \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{-2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{\frac {x \left (a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ y \left (x \right ) &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{-\frac {x \left (-a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.229 (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*}