Internal problem ID [8072]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 494.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 y x y^{\prime }+\left (-a^{2}+1\right ) x^{2}=0} \]
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 165
dsolve((y(x)^2-a^2*x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+(-a^2+1)*x^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \sqrt {a^{2}-1}\, x \\ y \left (x \right ) = -\sqrt {a^{2}-1}\, x \\ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}-\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-a^{4}+a^{2}}+\textit {\_a}}{\textit {\_a}^{4}-\textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}-a^{2}+1}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.279 (sec). Leaf size: 80
DSolve[(1 - a^2)*x^2 + 2*x*y[x]*y'[x] + (-(a^2*x^2) + y[x]^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a c_1-\sqrt {-x^2+c_1{}^2} \\ y(x)\to a c_1+\sqrt {-x^2+c_1{}^2} \\ y(x)\to -\sqrt {a^2-1} x \\ y(x)\to \sqrt {a^2-1} x \\ \end{align*}