Internal problem ID [7966]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 386.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.924 (sec). Leaf size: 27
dsolve(diff(y(x),x)^2+a*x^3*diff(y(x),x)-2*a*x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {a \,x^{4}}{8} \\ y \relax (x ) = c_{1} x^{2}+\frac {2 c_{1}^{2}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.902 (sec). Leaf size: 90
DSolve[-2*a*x^2*y[x] + a*x^3*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {i \sqrt {a} e^{2 c_1} x^2}{\sqrt {2}}-e^{4 c_1} \\ y(x)\to -\frac {i \sqrt {a} e^{2 c_1} x^2}{\sqrt {2}}-e^{4 c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}