28.1.29 problem 29

Internal problem ID [4335]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 29
Date solved : Monday, January 27, 2025 at 09:05:58 AM
CAS classification : [[_homogeneous, `class D`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y \left (2 x -y+2\right )+2 \left (x -y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 64

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

\begin{align*} y \left (x \right ) &= \frac {c_{1} x +\sqrt {{\mathrm e}^{x} c_{1} \left ({\mathrm e}^{x} c_{1} x^{2}+1\right )}\, {\mathrm e}^{-x}}{c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x -\sqrt {{\mathrm e}^{x} c_{1} \left ({\mathrm e}^{x} c_{1} x^{2}+1\right )}\, {\mathrm e}^{-x}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 42.485 (sec). Leaf size: 125

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

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