76.5.2 problem 2

Internal problem ID [17351]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 2
Date solved : Monday, March 31, 2025 at 03:55:49 PM
CAS classification : [_separable]

\begin{align*} \left (y^{4}+1\right ) y^{\prime }&=x^{4}+1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=(y(x)^4+1)*diff(y(x),x) = x^4+1; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {x^{5}}{5}+x -\frac {y^{5}}{5}-y+c_1 = 0 \]

✓ Mathematica. Time used: 4.516 (sec). Leaf size: 141

ode=(y[x]^4+1)*D[y[x],x]==x^4+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}-x^5-5 x-5 c_1\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}-x^5-5 x-5 c_1\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}-x^5-5 x-5 c_1\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}-x^5-5 x-5 c_1\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}-x^5-5 x-5 c_1\&,5\right ] \\ \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 + (y(x)**4 + 1)*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x^{5}}{5} - x + \frac {y^{5}{\left (x \right )}}{5} + y{\left (x \right )} = C_{1} \]