Internal problem ID [3721]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 997.
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}-3 a^{2} x y y^{\prime }-a^{2} x^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 195
dsolve((-a^2+1)*y(x)^2*diff(y(x),x)^2-3*a^2*x*y(x)*diff(y(x),x)-a^2*x^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (-2 \ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} \right )+2 c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {\left (-2 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}-3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.239 (sec). Leaf size: 342
DSolve[(1-a^2)y[x]^2 (y'[x])^2 -2 a^2 x y[x] y'[x]-a^2 x y[x] y'[x]-a^2 x^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\log \left (-\left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )\right )+\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}+\frac {2 y(x)^2}{x^2}\right )-\frac {2 \text {ArcTan}\left (\frac {1-\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}}{\sqrt {-5 a^4+2 a^2-1}}\right )}{\sqrt {-5 a^4+2 a^2-1}}}{4 a^2-4}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\log \left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )+\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}-\frac {2 y(x)^2}{x^2}\right )-\frac {2 \text {ArcTan}\left (\frac {\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}+1}{\sqrt {-5 a^4+2 a^2-1}}\right )}{\sqrt {-5 a^4+2 a^2-1}}}{4 a^2-4}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ] \\ \end{align*}