63.9.11 problem 1(k)

Internal problem ID [13067]
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 2.3.1 Nonhomogeneous Equations: Undetermined Coefficients. Exercises page 110
Problem number : 1(k)
Date solved : Monday, March 31, 2025 at 07:33:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=t^{3}+1-4 t \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 52

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = t^3+1-4*t*cos(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +\left (-4 t +8\right ) \sin \left (t \right )+t^{3}-3 t^{2}-4 \cos \left (t \right )+7 \]

✓ Mathematica. Time used: 0.55 (sec). Leaf size: 160

ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==t^3+1-4*t*Cos[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-t/2} \left (\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-\frac {2 e^{\frac {K[2]}{2}} \left (K[2]^3-4 \cos (K[2]) K[2]+1\right ) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t\frac {2 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \left (K[1]^3-4 \cos (K[1]) K[1]+1\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**3 + 4*t*cos(t) + x(t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t^{3} - 3 t^{2} - 4 t \sin {\left (t \right )} + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + 8 \sin {\left (t \right )} - 4 \cos {\left (t \right )} + 7 \]