Internal problem ID [3446]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 708.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (a^{2} x^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime }-a^{2} x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 197
dsolve((a^2*x^2+(x^2+y(x)^2)^2)*diff(y(x),x) = a^2*x*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1}\right ) x^{2}+\left (a^{2}+c_{1}\right )^{2}}-2 c_{1}}}{2} \\ y \relax (x ) = \frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1}\right ) x^{2}+\left (a^{2}+c_{1}\right )^{2}}-2 c_{1}}}{2} \\ y \relax (x ) = -\frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1}\right ) x^{2}+\left (a^{2}+c_{1}\right )^{2}}-2 c_{1}}}{2} \\ y \relax (x ) = \frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1}\right ) x^{2}+\left (a^{2}+c_{1}\right )^{2}}-2 c_{1}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.277 (sec). Leaf size: 272
DSolve[(a^2 x^2+(x^2+y[x]^2)^2)y'[x]==a^2 x y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}