Internal problem ID [3652]
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]
Solve \begin {gather*} \boxed {a \,x^{2} \left (y^{\prime }\right )^{2}-2 a x y y^{\prime }+a \left (-a +1\right ) x^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 123
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 \relax (x ) = \RootOf \left (-\ln \relax (x )-\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 \relax (x ) = \RootOf \left (-\ln \relax (x )+\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 \\ y \relax (x ) = c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.625 (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*}