6.5.20 problem 17

Internal problem ID [1644]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 04:41:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 58

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

\begin{align*} y &= \left (4 \ln \left (x \right )+c_1 \right )^{{1}/{4}} x \\ y &= -\left (4 \ln \left (x \right )+c_1 \right )^{{1}/{4}} x \\ y &= -i \left (4 \ln \left (x \right )+c_1 \right )^{{1}/{4}} x \\ y &= i \left (4 \ln \left (x \right )+c_1 \right )^{{1}/{4}} x \\ \end{align*}

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 76

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

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

✓ Sympy. Time used: 1.108 (sec). Leaf size: 68

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

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