Internal problem ID [3711]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 987.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (\left (-a^{2}+1\right ) x^{2}+y^{2}\right ) \left (y^{\prime }\right )^{2}+2 a^{2} x y y^{\prime }+x^{2}+\left (-a^{2}+1\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.302 (sec). Leaf size: 71
dsolve(((-a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2+2*a^2*x*y(x)*diff(y(x),x)+x^2+(-a^2+1)*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \tan \left (\RootOf \left (-2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 c_{1}\right )\right ) x \\ y \relax (x ) = \tan \left (\RootOf \left (2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 c_{1}\right )\right ) x \\ y \relax (x ) = c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.365 (sec). Leaf size: 115
DSolve[((1-a^2)x^2+y[x]^2) (y'[x])^2 +2 a^2 x y[x] y'[x]+x^2+(1-a^2) y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \text {ArcTan}\left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \text {ArcTan}\left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ] \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}