60.1.175 problem 176

Internal problem ID [10189]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 176
Date solved : Monday, January 27, 2025 at 06:34:26 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 1.179 (sec). Leaf size: 91

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