70.19.23 problem 23
Internal
problem
ID
[19033]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
5.
The
Laplace
transform.
Section
5.4
(Solving
differential
equations
with
Laplace
transform).
Problems
at
page
327
Problem
number
:
23
Date
solved
:
Thursday, October 02, 2025 at 03:37:07 PM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right )+\sin \left (t \right ) \end{align*}
With initial conditions
\begin{align*}
y_{1} \left (0\right )&=0 \\
y_{2} \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.276 (sec). Leaf size: 49
ode:=[diff(y__1(t),t) = -2*y__1(t)+y__2(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)+sin(t)];
ic:=[y__1(0) = 0, y__2(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
y_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-3 t}}{20}+\frac {{\mathrm e}^{-t}}{4}-\frac {\cos \left (t \right )}{5}+\frac {\sin \left (t \right )}{10} \\
y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t}}{20}+\frac {{\mathrm e}^{-t}}{4}+\frac {2 \sin \left (t \right )}{5}-\frac {3 \cos \left (t \right )}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.053 (sec). Leaf size: 294
ode={D[y1[t],t]==-2*y1[t]+1*y2[t]+0,D[y2[t],t]==1*y1[t]-2*y2[t]+Sin[t]};
ic={y1[0]==0,y2[0]==0};
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {y1}(t)&\to \frac {1}{2} e^{-3 t} \left (-\left (\left (e^{2 t}+1\right ) \int _1^0-\frac {1}{2} e^{K[1]} \left (-1+e^{2 K[1]}\right ) \sin (K[1])dK[1]\right )+\left (e^{2 t}+1\right ) \int _1^t-\frac {1}{2} e^{K[1]} \left (-1+e^{2 K[1]}\right ) \sin (K[1])dK[1]-\left (e^{2 t}-1\right ) \left (\int _1^0\frac {1}{2} e^{K[2]} \left (1+e^{2 K[2]}\right ) \sin (K[2])dK[2]-\int _1^t\frac {1}{2} e^{K[2]} \left (1+e^{2 K[2]}\right ) \sin (K[2])dK[2]\right )\right )\\ \text {y2}(t)&\to \frac {1}{2} e^{-3 t} \left (-\left (\left (e^{2 t}-1\right ) \int _1^0-\frac {1}{2} e^{K[1]} \left (-1+e^{2 K[1]}\right ) \sin (K[1])dK[1]\right )+\left (e^{2 t}-1\right ) \int _1^t-\frac {1}{2} e^{K[1]} \left (-1+e^{2 K[1]}\right ) \sin (K[1])dK[1]-\left (e^{2 t}+1\right ) \left (\int _1^0\frac {1}{2} e^{K[2]} \left (1+e^{2 K[2]}\right ) \sin (K[2])dK[2]-\int _1^t\frac {1}{2} e^{K[2]} \left (1+e^{2 K[2]}\right ) \sin (K[2])dK[2]\right )\right ) \end{align*}
✓ Sympy. Time used: 0.132 (sec). Leaf size: 51
from sympy import *
t = symbols("t")
y__1 = Function("y__1")
y__2 = Function("y__2")
ode=[Eq(2*y__1(t) - y__2(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 2*y__2(t) - sin(t) + Derivative(y__2(t), t),0)]
ics = {}
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
\[
\left [ y^{1}{\left (t \right )} = C_{1} e^{- t} - C_{2} e^{- 3 t} + \frac {\sin {\left (t \right )}}{10} - \frac {\cos {\left (t \right )}}{5}, \ y^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- 3 t} + \frac {2 \sin {\left (t \right )}}{5} - \frac {3 \cos {\left (t \right )}}{10}\right ]
\]