28.1.27 problem 27

Internal problem ID [4333]
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 : 27
Date solved : Monday, January 27, 2025 at 09:04:42 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.445 (sec). Leaf size: 53

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

\[ y \left (x \right ) = x \left (-1+\sqrt {2}\, \tan \left (\operatorname {RootOf}\left (\sqrt {2}\, \ln \left (2\right )+2 \sqrt {2}\, \ln \left (-\sec \left (\textit {\_Z} \right )^{2} \left (\sqrt {2}-2 \tan \left (\textit {\_Z} \right )\right ) x^{3}\right )+6 \sqrt {2}\, c_{1} +4 \textit {\_Z} \right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.195 (sec). Leaf size: 62

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

\[ \text {Solve}\left [\frac {1}{3} \left (\sqrt {2} \arctan \left (\frac {\frac {y(x)}{x}+1}{\sqrt {2}}\right )+\log \left (\frac {y(x)^2}{x^2}+\frac {2 y(x)}{x}+3\right )+\log \left (\frac {y(x)}{x}\right )\right )=-\log (x)+c_1,y(x)\right ] \]