76.2.8 problem 8

Internal problem ID [17273]
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 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 8
Date solved : Monday, March 31, 2025 at 03:48:18 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

ode:=(t^2+1)*diff(y(t),t)+4*t*y(t) = 1/(t^2+1)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\arctan \left (t \right )+c_1}{\left (t^{2}+1\right )^{2}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 31

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

\[ y(t)\to \frac {\int _1^t\frac {1}{K[1]^2+1}dK[1]+c_1}{\left (t^2+1\right )^2} \]

✓ Sympy. Time used: 0.443 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = \frac {C_{1} - i \log {\left (t - i \right )} + i \log {\left (t + i \right )}}{2 \left (t^{4} + 2 t^{2} + 1\right )} \]