7.24 problem 25

Internal problem ID [1084]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page 91
Problem number: 25.
ODE order: 1.
ODE degree: 1.

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

Solve \begin {gather*} \boxed {3 y x +2 y^{2}+y+\left (x^{2}+2 y x +x +2 y\right ) y^{\prime }=0} \end {gather*}

Solution by Maple

Time used: 0.047 (sec). Leaf size: 103

dsolve((3*x*y(x)+2*y(x)^2+y(x))+(x^2+2*x*y(x)+x+2*y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 14.459 (sec). Leaf size: 105

DSolve[(3*x*y[x]+2*y[x]^2+y[x])+(x^2+2*x*y[x]+x+2*y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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