Internal problem ID [4161]
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`], _rational, _dAlembert]
\[ \boxed {\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +y^{2}=x^{2} a^{2}} \]
✓ Solution by Maple
Time used: 0.062 (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 \left (x \right )-\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {-\frac {a^{2} x^{2}-x^{2}-y \left (x \right )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{a}-c_{1} = 0 \ln \left (x \right )+\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {-\frac {a^{2} x^{2}-x^{2}-y \left (x \right )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{a}-c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.04 (sec). Leaf size: 223
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 {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \end{align*}