63.9.21 problem 4

Internal problem ID [13148]
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 : 4
Date solved : Tuesday, January 28, 2025 at 05:09:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }-3 x^{\prime }-40 x&=2 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 24

dsolve([diff(x(t),t$2)-3*diff(x(t),t)-40*x(t)=2*exp(-t),x(0) = 0, D(x)(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (22 \,{\mathrm e}^{13 t}-13 \,{\mathrm e}^{4 t}-9\right ) {\mathrm e}^{-5 t}}{234} \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 30

DSolve[{D[x[t],{t,2}]-3*D[x[t],t]-40*x[t]==2*Exp[-t],{x[0]==0,Derivative[1][x][0 ]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{234} e^{-5 t} \left (-13 e^{4 t}+22 e^{13 t}-9\right ) \]