73.7.33 problem 33

Internal problem ID [15112]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 33
Date solved : Thursday, March 13, 2025 at 05:40:35 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.551 (sec). Leaf size: 96

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

\begin{align*} y(x)\to -\sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to -i \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to i \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ \end{align*}

✓ Sympy. Time used: 1.718 (sec). Leaf size: 68

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

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