29.13.9 problem 363

Internal problem ID [4963]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 363
Date solved : Monday, January 27, 2025 at 09:56:02 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x \left (-x^{2}+1\right ) y^{\prime }+x^{2}+\left (-x^{2}+1\right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 33

dsolve(x*(-x^2+1)*diff(y(x),x)+x^2+(-x^2+1)*y(x)^2 = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-c_{1} \operatorname {EllipticCE}\left (x \right )+\operatorname {EllipticK}\left (x \right )-\operatorname {EllipticE}\left (x \right )}{c_{1} \operatorname {EllipticCK}\left (x \right )-c_{1} \operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticE}\left (x \right )} \]

✓ Solution by Mathematica

Time used: 0.963 (sec). Leaf size: 91

DSolve[x(1-x^2)D[y[x],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| \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*}