28.1.70 problem 73

Internal problem ID [4376]
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 : 73
Date solved : Sunday, March 30, 2025 at 03:10:51 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime }&=\frac {4 x^{3} y^{2}}{x^{4} y+2} \end{align*}

✓ Maple. Time used: 0.251 (sec). Leaf size: 45

ode:=diff(y(x),x) = 4*x^3*y(x)^2/(x^4*y(x)+2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{4}-\sqrt {x^{8}+4 c_1}}{2 c_1} \\ y &= \frac {x^{4}+\sqrt {x^{8}+4 c_1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.422 (sec). Leaf size: 56

ode=D[y[x],x]==(4*x^3*y[x]^2)/(x^4*y[x]+2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {2}{-x^4+\sqrt {x^8+4 c_1}} \\ y(x)\to -\frac {2}{x^4+\sqrt {x^8+4 c_1}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.169 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**3*y(x)**2/(x**4*y(x) + 2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1} x^{4}}{2} - \frac {\sqrt {C_{1} \left (C_{1} x^{8} + 4\right )}}{2}, \ y{\left (x \right )} = \frac {C_{1} x^{4}}{2} + \frac {\sqrt {C_{1} \left (C_{1} x^{8} + 4\right )}}{2}\right ] \]