Internal problem ID [8033]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 453.
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 ) x^{2} \left (y^{\prime }\right )^{2}+2 x y y^{\prime }-y^{2}+a^{2} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 2.422 (sec). Leaf size: 229
dsolve((a^2-1)*x^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-y(x)^2+a^2*x^2 = 0,y(x), singsol=all)
\begin{align*} \ln \relax (x )-\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \relax (x )}{\sqrt {-a^{2}}\, \sqrt {-\frac {x^{2} a^{2}-x^{2}-y \relax (x )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \relax (x )^{2}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {\sqrt {\frac {-x^{2} a^{2}+x^{2}+y \relax (x )^{2}}{x^{2}}}\, x +y \relax (x )}{x}\right )}{a}-c_{1} = 0 \\ \ln \relax (x )+\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \relax (x )}{\sqrt {-a^{2}}\, \sqrt {-\frac {x^{2} a^{2}-x^{2}-y \relax (x )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \relax (x )^{2}}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {-x^{2} a^{2}+x^{2}+y \relax (x )^{2}}{x^{2}}}\, x +y \relax (x )}{x}\right )}{a}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.817 (sec). Leaf size: 283
DSolve[a^2*x^2 - y[x]^2 + 2*x*y[x]*y'[x] + (-1 + a^2)*x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {a \left (\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-a-\frac {i y(x)}{x}+1\right )+\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a-\frac {i y(x)}{x}-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-\frac {i y(x)}{x}\right )\right )}{a^2-1}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {a \left (\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-a-\frac {i y(x)}{x}-1\right )+\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a-\frac {i y(x)}{x}+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-\frac {i y(x)}{x}\right )\right )}{a^2-1}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}