29.22.4 problem 610

Internal problem ID [5204]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 22
Problem number : 610
Date solved : Tuesday, January 28, 2025 at 02:41:33 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 110

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

\begin{align*} y \left (x \right ) &= \frac {x^{2}-c_{1} x +\sqrt {x^{4}-2 c_{1} x^{3}+\left (c_{1}^{2}-2\right ) x^{2}+\left (2 c_{1} +4\right ) x -4 c_{1} +1}-1}{-2 x +2 c_{1}} \\ y \left (x \right ) &= \frac {-x^{2}+c_{1} x +\sqrt {x^{4}-2 c_{1} x^{3}+\left (c_{1}^{2}-2\right ) x^{2}+\left (2 c_{1} +4\right ) x -4 c_{1} +1}+1}{2 x -2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.899 (sec). Leaf size: 146

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

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