Internal problem ID [8034]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 454.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {a \,x^{2} \left (y^{\prime }\right )^{2}-2 a x y y^{\prime }+y^{2}-a \left (a -1\right ) x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.333 (sec). Leaf size: 123
dsolve(a*x^2*diff(y(x),x)^2-2*a*x*y(x)*diff(y(x),x)+y(x)^2-a*(a-1)*x^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a \right ) a}}{a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a}d \textit {\_a} \right )+c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a \right ) a}}{a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.639 (sec). Leaf size: 241
DSolve[-((-1 + a)*a*x^2) + y[x]^2 - 2*a*x*y[x]*y'[x] + a*x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (x^{2 \sqrt {\frac {a-1}{a}}}-e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) \\ y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) \\ y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) \\ \end{align*}