63.12.2 problem 1(b)

Internal problem ID [13164]
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(b)
Date solved : Tuesday, January 28, 2025 at 05:11:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 25

dsolve(diff(x(t),t$2)-x(t)=t*exp(t),x(t), singsol=all)
 

\[ x \left (t \right ) = c_{2} {\mathrm e}^{-t}+\frac {{\mathrm e}^{t} \left (t^{2}+4 c_{1} -t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 54

DSolve[D[x[t],{t,2}]-x[t]==t*Exp[t],x[t],t,IncludeSingularSolutions -> True]
 

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