28.1.90 problem 111

Internal problem ID [4396]
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 : 111
Date solved : Monday, January 27, 2025 at 09:14:12 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 72

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

\begin{align*} y \left (x \right ) &= \frac {\left (12 x^{2}+8 c_{1} \right )^{{1}/{3}}}{2 x} \\ y \left (x \right ) &= -\frac {\left (12 x^{2}+8 c_{1} \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y \left (x \right ) &= \frac {\left (12 x^{2}+8 c_{1} \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.244 (sec). Leaf size: 80

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

\begin{align*} y(x)\to -\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{3 x^2+2 c_1}}{x} \\ y(x)\to \frac {\sqrt [3]{\frac {3 x^2}{2}+c_1}}{x} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {3 x^2}{2}+c_1}}{x} \\ \end{align*}