Internal problem ID [3111]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 13
Problem number: 363.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x \left (1-x^{2}\right ) y^{\prime }+x^{2}+\left (1-x^{2}\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 45
dsolve(x*(-x^2+1)*diff(y(x),x)+x^2+(-x^2+1)*y(x)^2 = 0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\EllipticK \relax (x )}{c_{1} \EllipticCE \relax (x )-c_{1} \EllipticCK \relax (x )+\EllipticE \relax (x )}+\frac {c_{1} \EllipticCE \relax (x )+\EllipticE \relax (x )}{c_{1} \EllipticCE \relax (x )-c_{1} \EllipticCK \relax (x )+\EllipticE \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.521 (sec). Leaf size: 91
DSolve[x(1-x^2)y'[x]+x^2+(1-x^2)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \left (\pi G_{2,2}^{2,0}\left (x^2| {c} \frac {1}{2},\frac {3}{2} \\ 0,1 \\ \\ \right )+c_1 \left (K\left (x^2\right )-E\left (x^2\right )\right )\right )}{\pi G_{2,2}^{2,0}\left (x^2| {c} \frac {1}{2},\frac {3}{2} \\ 0,0 \\ \\ \right )+2 c_1 E\left (x^2\right )} \\ y(x)\to 1-\frac {K\left (x^2\right )}{E\left (x^2\right )} \\ \end{align*}