1.235 problem 236

Internal problem ID [7816]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 236.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]

Solve \begin {gather*} \boxed {x \left (4+y\right ) y^{\prime }-2 x -2 y-y^{2}=0} \end {gather*}

Solution by Maple

Time used: 0.002 (sec). Leaf size: 145

dsolve(x*(y(x)+4)*diff(y(x),x)-y(x)^2-2*y(x)-2*x=0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \frac {\left (x +4\right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{x +4}}\, x +4 x^{\frac {3}{2}}+16 \sqrt {x}}{\left (x +4\right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{x +4}}-x^{\frac {3}{2}}-4 \sqrt {x}} \\ y \relax (x ) = \frac {\left (x +4\right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{x +4}}\, x -4 x^{\frac {3}{2}}-16 \sqrt {x}}{\left (x +4\right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{x +4}}+x^{\frac {3}{2}}+4 \sqrt {x}} \\ \end{align*}

Solution by Mathematica

Time used: 0.823 (sec). Leaf size: 90

DSolve[x*(y[x]+4)*y'[x]-y[x]^2-2*y[x]-2*x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {c_1 x+\sqrt {x} \sqrt {x+4} \sqrt {-\frac {4}{x+4}+c_1}}{-1+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*}