8.26.8 problem 8
Internal
problem
ID
[2781]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
3.
Systems
of
differential
equations.
Section
3.13
(Solving
systems
by
Laplace
transform).
Page
370
Problem
number
:
8
Date
solved
:
Tuesday, September 30, 2025 at 05:52:37 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )+f_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+f_{2} \left (t \right ) \end{align*}
With initial conditions
\begin{align*}
x_{1} \left (0\right )&=0 \\
x_{2} \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.527 (sec). Leaf size: 100
ode:=[diff(x__1(t),t) = x__2(t)+f__1(t), diff(x__2(t),t) = -x__1(t)+f__2(t)];
ic:=[x__1(0) = 0, x__2(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
x_{1} \left (t \right ) &= \sin \left (t \right ) f_{1} \left (0\right )+\int _{0}^{t}\cos \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}f_{1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \sin \left (t \right )-\int _{0}^{t}\sin \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}f_{1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \cos \left (t \right ) \\
x_{2} \left (t \right ) &= \cos \left (t \right ) f_{1} \left (0\right )+\int _{0}^{t}\cos \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}f_{1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \cos \left (t \right )+\int _{0}^{t}\sin \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}f_{1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \sin \left (t \right )-f_{1} \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 216
ode={D[x1[t],t]==0*x1[t]+1*x2[t]+f1[t],D[x2[t],t]==-1*x1[t]-0*x2[t]+f2[t]};
ic={x1[0]==0,x2[0]==0};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to -\cos (t) \int _1^0(\cos (K[1]) \text {f1}(K[1])-\text {f2}(K[1]) \sin (K[1]))dK[1]+\cos (t) \int _1^t(\cos (K[1]) \text {f1}(K[1])-\text {f2}(K[1]) \sin (K[1]))dK[1]+\sin (t) \left (\int _1^t(\cos (K[2]) \text {f2}(K[2])+\text {f1}(K[2]) \sin (K[2]))dK[2]-\int _1^0(\cos (K[2]) \text {f2}(K[2])+\text {f1}(K[2]) \sin (K[2]))dK[2]\right )\\ \text {x2}(t)&\to \sin (t) \int _1^0(\cos (K[1]) \text {f1}(K[1])-\text {f2}(K[1]) \sin (K[1]))dK[1]-\sin (t) \int _1^t(\cos (K[1]) \text {f1}(K[1])-\text {f2}(K[1]) \sin (K[1]))dK[1]+\cos (t) \left (\int _1^t(\cos (K[2]) \text {f2}(K[2])+\text {f1}(K[2]) \sin (K[2]))dK[2]-\int _1^0(\cos (K[2]) \text {f2}(K[2])+\text {f1}(K[2]) \sin (K[2]))dK[2]\right ) \end{align*}
✓ Sympy. Time used: 0.232 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-f__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(-f__2(t) + x__1(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + \sin {\left (t \right )} \int \left (f^{1}{\left (t \right )} \sin {\left (t \right )} + f^{2}{\left (t \right )} \cos {\left (t \right )}\right )\, dt + \cos {\left (t \right )} \int \left (f^{1}{\left (t \right )} \cos {\left (t \right )} - f^{2}{\left (t \right )} \sin {\left (t \right )}\right )\, dt, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} - \sin {\left (t \right )} \int \left (f^{1}{\left (t \right )} \cos {\left (t \right )} - f^{2}{\left (t \right )} \sin {\left (t \right )}\right )\, dt + \cos {\left (t \right )} \int \left (f^{1}{\left (t \right )} \sin {\left (t \right )} + f^{2}{\left (t \right )} \cos {\left (t \right )}\right )\, dt\right ]
\]