57.9.7 problem 1(g)
Internal
problem
ID
[14415]
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(g)
Date
solved
:
Thursday, October 02, 2025 at 09:36:54 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{\prime \prime }+x^{\prime }+x&=t \,{\mathrm e}^{-t} \sin \left (\pi t \right ) \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 133
ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = t*exp(-t)*sin(Pi*t);
dsolve(ode,x(t), singsol=all);
\[
x = -\frac {{\mathrm e}^{-t} \left (-c_1 \,{\mathrm e}^{\frac {t}{2}} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \cos \left (\frac {\sqrt {3}\, t}{2}\right )-c_2 \,{\mathrm e}^{\frac {t}{2}} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\left (\pi ^{6} t +\left (-2 t +3\right ) \pi ^{4}+\left (2 t -1\right ) \pi ^{2}-t -1\right ) \sin \left (\pi t \right )-\cos \left (\pi t \right ) \left (\left (t -2\right ) \pi ^{4}+\left (-t +4\right ) \pi ^{2}+t \right ) \pi \right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}}
\]
✓ Mathematica. Time used: 0.045 (sec). Leaf size: 123
ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==t*Exp[-t]*Sin[Pi*t];
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to e^{-t} \left (\frac {\left (\pi ^2 (1-2 t)-\pi ^6 t+t+\pi ^4 (2 t-3)+1\right ) \sin (\pi t)+\pi \left (-\pi ^2 (t-4)+\pi ^4 (t-2)+t\right ) \cos (\pi t)}{\left (1-\pi ^2+\pi ^4\right )^2}+c_2 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.354 (sec). Leaf size: 231
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(-t*exp(-t)*sin(pi*t) + x(t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \left (- \frac {\pi ^{2} t \sin {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} + \frac {t \sin {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} + \frac {\pi t \cos {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} - \frac {3 \pi ^{4} \sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {\sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {\pi ^{2} \sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} - \frac {2 \pi ^{5} \cos {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {4 \pi ^{3} \cos {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}}\right ) e^{- t}
\]