60.1.278 problem 279

Internal problem ID [10292]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 279
Date solved : Tuesday, January 28, 2025 at 04:36:04 PM
CAS classification : [_rational]

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

✓ Solution by Maple

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

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

\begin{align*} y &= \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 &= \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 c_{1} +2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.941 (sec). Leaf size: 146

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