60.1.177 problem 178

Internal problem ID [10191]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 178
Date solved : Monday, January 27, 2025 at 06:36:19 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y+x^{2}-3&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 103

dsolve(2*x*(x^2-1)*diff(y(x),x) + 2*(x^2-1)*y(x)^2 - (3*x^2-5)*y(x) + x^2 - 3=0,y(x), singsol=all)
 

\[ y = \frac {-2 \sqrt {2}\, \sqrt {x +1}\, \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {1-x}+\sqrt {x -1}\, \sqrt {x}\, \sqrt {x +1}\, c_{1} -2 x}{\sqrt {x +1}\, \left (-2 \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2}\, \sqrt {1-x}+c_{1} \sqrt {x -1}\, \sqrt {x}\right )} \]

✓ Solution by Mathematica

Time used: 0.372 (sec). Leaf size: 174

DSolve[2*x*(x^2-1)*D[y[x],x] + 2*(x^2-1)*y[x]^2 - (3*x^2-5)*y[x] + x^2 - 3==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 1+\frac {\exp \left (\int _1^x\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{-\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{K[2]}dK[2]+c_1} \\ y(x)\to 1 \\ y(x)\to 1-\frac {\exp \left (\int _1^x\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{K[2]}dK[2]} \\ \end{align*}