Internal problem ID [3653]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 926.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {\left (-a^{2}+1\right ) x^{2} \left (y^{\prime }\right )^{2}-2 y^{\prime } y x -a^{2} x^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 3.156 (sec). Leaf size: 229
dsolve((-a^2+1)*x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-a^2*x^2+y(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 {a^{2} x^{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 {-a^{2} x^{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 {a^{2} x^{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 {-a^{2} x^{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: 1.565 (sec). Leaf size: 395
DSolve[(1-a^2)x^2 (y'[x])^2-2 x y[x] y'[x]-a^2 x^2 + y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {a \left (-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )+\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \text {ArcTan}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 i \text {ArcTan}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}