29.24.3 problem 665

Internal problem ID [5256]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 665
Date solved : Monday, January 27, 2025 at 10:49:56 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.229 (sec). Leaf size: 137

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {x c_{1} \left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right )}}{2 c_{1} x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {x c_{1} \left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right )}}{2 c_{1} x} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {x c_{1} \left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right )}}{2 c_{1} x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {x c_{1} \left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right )}}{2 c_{1} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 65

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

\begin{align*} y(x)\to \frac {1}{2} \left (c_1-\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}\right ) \\ y(x)\to \frac {1}{2} \left (\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}+c_1\right ) \\ y(x)\to 0 \\ \end{align*}