63.9.6 problem 1(f)

Internal problem ID [13054]
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(f)
Date solved : Wednesday, March 05, 2025 at 09:02:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&={\mathrm e}^{2 t} \cos \left (t \right )+t^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 52

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = exp(2*t)*cos(t)+t^2; 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +t^{2}-2 t +\frac {\left (6 \cos \left (t \right )+5 \sin \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 164

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

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

✓ Sympy. Time used: 0.306 (sec). Leaf size: 54

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

\[ x{\left (t \right )} = t^{2} - 2 t + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \frac {\left (5 \sin {\left (t \right )} + 6 \cos {\left (t \right )}\right ) e^{2 t}}{61} \]