4.20 problem 20

Internal problem ID [5448]

Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number: 20.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _exact, _rational]

Solve \begin {gather*} \boxed {\frac {-x y^{\prime }+y}{\left (x +y\right )^{2}}+y^{\prime }-1=0} \end {gather*}

Solution by Maple

Time used: 0.006 (sec). Leaf size: 65

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

\begin{align*} y \relax (x ) = \frac {c_{1}}{4}+\frac {1}{4}-\frac {\sqrt {c_{1}^{2}+8 c_{1} x +16 x^{2}+2 c_{1}-8 x +1}}{4} \\ y \relax (x ) = \frac {c_{1}}{4}+\frac {1}{4}+\frac {\sqrt {c_{1}^{2}+8 c_{1} x +16 x^{2}+2 c_{1}-8 x +1}}{4} \\ \end{align*}

Solution by Mathematica

Time used: 0.359 (sec). Leaf size: 70

DSolve[(y[x]-x*y'[x])/(x+y[x])^2+y'[x]==1,y[x],x,IncludeSingularSolutions -> True]
 

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