Internal problem ID [4160]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 925.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+y^{2}=-a \left (-a +1\right ) x^{2}} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 138
dsolve(a*x^2*diff(y(x),x)^2-2*a*x*y(x)*diff(y(x),x)+a*(1-a)*x^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = x \sqrt {-a} y \left (x \right ) = -x \sqrt {-a} y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a \right ) a}}{a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a}d \textit {\_a} \right )+c_{1} \right ) x y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a \right ) a}}{a \,\textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-a}d \textit {\_a} +c_{1} \right ) x \end{align*}
✓ Solution by Mathematica
Time used: 0.637 (sec). Leaf size: 241
DSolve[a x^2 (y'[x])^2-2 a x y[x] y'[x]+a(1-a)x^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (x^{2 \sqrt {\frac {a-1}{a}}}-e^{2 c_1}\right ) y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) \end{align*}