28.1.12 problem 12

Internal problem ID [4318]
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 : 12
Date solved : Monday, January 27, 2025 at 09:02:10 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.955 (sec). Leaf size: 221

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

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

✓ Solution by Mathematica

Time used: 8.613 (sec). Leaf size: 218

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

\begin{align*} y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to -\sqrt {\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {\sqrt {x^4}-x^2} \\ \end{align*}