63.9.20 problem 3
Internal
problem
ID
[13068]
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
:
3
Date
solved
:
Wednesday, March 05, 2025 at 09:13:14 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{\prime \prime }-b x^{\prime }+x&=\sin \left (2 t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right )&=0\\ x^{\prime }\left (0\right )&=0 \end{align*}
✓ Maple. Time used: 0.228 (sec). Leaf size: 135
ode:=diff(diff(x(t),t),t)-b*diff(x(t),t)+x(t) = sin(2*t);
ic:=x(0) = 0, D(x)(0) = 0;
dsolve([ode,ic],x(t), singsol=all);
\[
x \left (t \right ) = \frac {\left (-\sqrt {b^{2}-4}\, b^{2}-b^{3}-6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{-\frac {\left (-b +\sqrt {b^{2}-4}\right ) t}{2}}+\left (\sqrt {b^{2}-4}\, b^{2}-b^{3}+6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{\frac {\left (b +\sqrt {b^{2}-4}\right ) t}{2}}+2 \left (b^{3}-4 b \right ) \cos \left (2 t \right )+3 \left (-b^{2}+4\right ) \sin \left (2 t \right )}{4 b^{4}-7 b^{2}-36}
\]
✓ Mathematica. Time used: 0.339 (sec). Leaf size: 241
ode=D[x[t],{t,2}]-b*D[x[t],t]+x[t]==Sin[2*t];
ic={x[0]==0,Derivative[1][x][0 ]==0};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\[
x(t)\to -e^{\frac {1}{2} \left (b-\sqrt {b^2-4}\right ) t} \int _1^0-\frac {e^{\frac {1}{2} \left (\sqrt {b^2-4}-b\right ) K[1]} \sin (2 K[1])}{\sqrt {b^2-4}}dK[1]+e^{\frac {1}{2} \left (b-\sqrt {b^2-4}\right ) t} \int _1^t-\frac {e^{\frac {1}{2} \left (\sqrt {b^2-4}-b\right ) K[1]} \sin (2 K[1])}{\sqrt {b^2-4}}dK[1]+e^{\frac {1}{2} \left (\sqrt {b^2-4}+b\right ) t} \left (\int _1^t\frac {e^{-\frac {1}{2} \left (b+\sqrt {b^2-4}\right ) K[2]} \sin (2 K[2])}{\sqrt {b^2-4}}dK[2]-\int _1^0\frac {e^{-\frac {1}{2} \left (b+\sqrt {b^2-4}\right ) K[2]} \sin (2 K[2])}{\sqrt {b^2-4}}dK[2]\right )
\]
✓ Sympy. Time used: 0.365 (sec). Leaf size: 236
from sympy import *
t = symbols("t")
b = symbols("b")
x = Function("x")
ode = Eq(-b*Derivative(x(t), t) + x(t) - sin(2*t) + Derivative(x(t), (t, 2)),0)
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = \frac {2 b \cos {\left (2 t \right )}}{4 b^{2} + 9} + \left (- \frac {b^{2}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {b \sqrt {b^{2} - 4}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {6}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}}\right ) e^{\frac {t \left (b - \sqrt {b^{2} - 4}\right )}{2}} + \left (\frac {b^{2}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {b \sqrt {b^{2} - 4}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} + \frac {6}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}}\right ) e^{\frac {t \left (b + \sqrt {b^{2} - 4}\right )}{2}} - \frac {3 \sin {\left (2 t \right )}}{4 b^{2} + 9}
\]