44.5.28 problem 28

Internal problem ID [7090]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 28
Date solved : Sunday, March 30, 2025 at 11:40:23 AM
CAS classification : [_separable]

\begin{align*} \left (x^{4}+1\right ) y^{\prime }+x \left (1+4 y^{2}\right )&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 21

ode:=(x^4+1)*diff(y(x),x)+x*(1+4*y(x)^2) = 0; 
ic:=y(1) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {-x^{2}+1}{2 x^{2}+2} \]

✓ Mathematica. Time used: 0.296 (sec). Leaf size: 20

ode=(1+x^4)*D[y[x],x]+x*(1+4*y[x]^2)==0; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} \cot \left (\arctan \left (x^2\right )+\frac {\pi }{4}\right ) \]

✓ Sympy. Time used: 0.472 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(4*y(x)**2 + 1) + (x**4 + 1)*Derivative(y(x), x),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\cot {\left (\operatorname {atan}{\left (x^{2} \right )} + \frac {\pi }{4} \right )}}{2} \]