63.12.5 problem 1(e)

Internal problem ID [13096]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.2 Variation of parameters. Exercises page 124
Problem number : 1(e)
Date solved : Monday, March 31, 2025 at 07:34:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\frac {1}{1+t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(x(t),t),t)+x(t) = 1/(t+1); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +\operatorname {Ci}\left (1+t \right ) \sin \left (1+t \right )-\operatorname {Si}\left (1+t \right ) \cos \left (1+t \right ) \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 57

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

\[ x(t)\to \cos (t) \int _1^t-\frac {\sin (K[1])}{K[1]+1}dK[1]+\sin (t) \int _1^t\frac {\cos (K[2])}{K[2]+1}dK[2]+c_1 \cos (t)+c_2 \sin (t) \]

✓ Sympy. Time used: 1.432 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = \left (C_{1} - \int \frac {\sin {\left (t \right )}}{t + 1}\, dt\right ) \cos {\left (t \right )} + \left (C_{2} + \int \frac {\cos {\left (t \right )}}{t + 1}\, dt\right ) \sin {\left (t \right )} \]