Internal problem ID [8079]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 499.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (-a^{2}+1\right ) y^{2} \left (y^{\prime }\right )^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.315 (sec). Leaf size: 201
dsolve((-a^2+1)*y(x)^2*diff(y(x),x)^2-2*a^2*x*y(x)*diff(y(x),x)+y(x)^2-a^2*x^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x a}{\sqrt {-a^{2}+1}} \\ y \relax (x ) = -\frac {x a}{\sqrt {-a^{2}+1}} \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {\left (-\textit {\_a}^{2} a^{2}+\textit {\_a}^{2}-a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.133 (sec). Leaf size: 251
DSolve[-(a^2*x^2) + y[x]^2 - 2*a^2*x*y[x]*y'[x] + (1 - a^2)*y[x]^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ \end{align*}