Internal problem ID [8512]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 176.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x \left (x^{2}-1\right ) y^{\prime }+\left (x^{2}-1\right ) y^{2}=x^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 45
dsolve(x*(x^2-1)*diff(y(x),x) + (x^2-1)*y(x)^2 - x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\operatorname {EllipticK}\left (x \right )}{c_{1} \operatorname {EllipticCE}\left (x \right )-c_{1} \operatorname {EllipticCK}\left (x \right )+\operatorname {EllipticE}\left (x \right )}+\frac {c_{1} \operatorname {EllipticCE}\left (x \right )+\operatorname {EllipticE}\left (x \right )}{c_{1} \operatorname {EllipticCE}\left (x \right )-c_{1} \operatorname {EllipticCK}\left (x \right )+\operatorname {EllipticE}\left (x \right )} \]
✓ Solution by Mathematica
Time used: 0.9 (sec). Leaf size: 91
DSolve[x*(x^2-1)*y'[x] + (x^2-1)*y[x]^2 - 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| \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,1 \\ \end {array} \right )+c_1 \left (\operatorname {EllipticK}\left (x^2\right )-\operatorname {EllipticE}\left (x^2\right )\right )\right )}{\pi G_{2,2}^{2,0}\left (x^2| \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,0 \\ \end {array} \right )+2 c_1 \operatorname {EllipticE}\left (x^2\right )} y(x)\to 1-\frac {\operatorname {EllipticK}\left (x^2\right )}{\operatorname {EllipticE}\left (x^2\right )} \end{align*}