70.19.20 problem 20
Internal
problem
ID
[19030]
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
:
20
Date
solved
:
Thursday, October 02, 2025 at 03:37:06 PM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )-y_{2} \left (t \right )+2 \,{\mathrm e}^{t}\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right )+\sin \left (2 t \right ) \end{align*}
With initial conditions
\begin{align*}
y_{1} \left (0\right )&=1 \\
y_{2} \left (0\right )&=2 \\
\end{align*}
✓ Maple. Time used: 0.281 (sec). Leaf size: 67
ode:=[diff(y__1(t),t) = -4*y__1(t)-y__2(t)+2*exp(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)+sin(2*t)];
ic:=[y__1(0) = 1, y__2(0) = 2];
dsolve([ode,op(ic)]);
\begin{align*}
y_{1} \left (t \right ) &= \frac {749 \,{\mathrm e}^{-3 t}}{1352}-\frac {69 \,{\mathrm e}^{-3 t} t}{26}+\frac {3 \,{\mathrm e}^{t}}{8}+\frac {12 \cos \left (2 t \right )}{169}-\frac {5 \sin \left (2 t \right )}{169} \\
y_{2} \left (t \right ) &= \frac {2839 \,{\mathrm e}^{-3 t}}{1352}+\frac {69 \,{\mathrm e}^{-3 t} t}{26}+\frac {{\mathrm e}^{t}}{8}+\frac {44 \sin \left (2 t \right )}{169}-\frac {38 \cos \left (2 t \right )}{169} \\
\end{align*}
✓ Mathematica. Time used: 0.229 (sec). Leaf size: 392
ode={D[y1[t],t]==-4*y1[t]-1*y2[t]+2*Exp[t],D[y2[t],t]==1*y1[t]-2*y2[t]+Sin[2*t]};
ic={y1[0]==1,y2[0]==2};
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {y1}(t)&\to e^{-3 t} \left (t \int _1^0e^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]-t \int _1^te^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]+(t-1) \int _1^0e^{3 K[1]} \left (2 e^{K[1]} (K[1]+1)+K[1] \sin (2 K[1])\right )dK[1]-(t-1) \int _1^te^{3 K[1]} \left (2 e^{K[1]} (K[1]+1)+K[1] \sin (2 K[1])\right )dK[1]-3 t+1\right )\\ \text {y2}(t)&\to e^{-3 t} \left (t \left (-\int _1^0e^{3 K[1]} \left (2 e^{K[1]} (K[1]+1)+K[1] \sin (2 K[1])\right )dK[1]\right )+t \int _1^te^{3 K[1]} \left (2 e^{K[1]} (K[1]+1)+K[1] \sin (2 K[1])\right )dK[1]-t \int _1^0e^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]+t \int _1^te^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]+\int _1^te^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]-\int _1^0e^{3 K[2]} \left (-2 e^{K[2]} K[2]-(K[2]-1) \sin (2 K[2])\right )dK[2]+3 t+2\right ) \end{align*}
✓ Sympy. Time used: 1.165 (sec). Leaf size: 82
from sympy import *
t = symbols("t")
y__1 = Function("y__1")
y__2 = Function("y__2")
ode=[Eq(4*y__1(t) + y__2(t) - 2*exp(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 2*y__2(t) - sin(2*t) + Derivative(y__2(t), t),0)]
ics = {}
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
\[
\left [ y^{1}{\left (t \right )} = - C_{2} t e^{- 3 t} - \left (C_{1} - C_{2}\right ) e^{- 3 t} + \frac {3 e^{t}}{8} - \frac {5 \sin {\left (2 t \right )}}{169} + \frac {12 \cos {\left (2 t \right )}}{169}, \ y^{2}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} t e^{- 3 t} + \frac {e^{t}}{8} + \frac {44 \sin {\left (2 t \right )}}{169} - \frac {38 \cos {\left (2 t \right )}}{169}\right ]
\]