50.4.20 problem 20

Internal problem ID [7869]
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
Date solved : Monday, January 27, 2025 at 03:29:05 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 61

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

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

✓ Solution by Mathematica

Time used: 0.469 (sec). Leaf size: 76

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

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