22.1.43 problem 44

Internal problem ID [4349]
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 : 44
Date solved : Tuesday, September 30, 2025 at 07:22:14 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2}-\left (x y+x^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 35

ode:=y(x)^2-(x*y(x)+x^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.334 (sec). Leaf size: 67

ode=(y[x]^2)-(x*y[x]+x^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x^2 \left (1+\sqrt {\frac {1}{x^3}} \sqrt {x \left (x^2+c_1\right )}\right )\\ y(x)&\to x^2 \left (-1+\sqrt {\frac {1}{x^3}} \sqrt {x \left (x^2+c_1\right )}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.487 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x**3 - x*y(x))*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \left (- x - \sqrt {C_{1} + x^{2}}\right ), \ y{\left (x \right )} = x \left (- x + \sqrt {C_{1} + x^{2}}\right )\right ] \]