29.19.10 problem 523

Internal problem ID [5119]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 523
Date solved : Monday, January 27, 2025 at 10:12:14 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x \left (4+y\right ) y^{\prime }&=2 x +2 y+y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 121

dsolve(x*(4+y(x))*diff(y(x),x) = 2*x+2*y(x)+y(x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.185 (sec). Leaf size: 89

DSolve[x(4+y[x])D[y[x],x]==2 x+2 y[x]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -4+\frac {1}{\frac {1}{x+4}-\frac {\sqrt {x}}{(x+4)^{3/2} \sqrt {-\frac {4}{x+4}+c_1}}} \\ y(x)\to -4+\frac {1}{\frac {1}{x+4}+\frac {\sqrt {x}}{(x+4)^{3/2} \sqrt {-\frac {4}{x+4}+c_1}}} \\ y(x)\to x \\ \end{align*}