14.20.12 problem 12
Internal
problem
ID
[2709]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.14,
The
method
of
elimination
for
systems.
Excercises
page
258
Problem
number
:
12
Date
solved
:
Tuesday, March 04, 2025 at 02:40:09 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+\textit {f\_1} \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+f_{2} \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 1.849 (sec). Leaf size: 100
ode:=[diff(x(t),t) = y(t)+f_1(t), diff(y(t),t) = -x(t)+f__2(t)];
ic:=x(0) = 0y(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= \sin \left (t \right ) \textit {f\_1} \left (0\right )+\left (\int _{0}^{t}\cos \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}\textit {f\_1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right ) \sin \left (t \right )-\left (\int _{0}^{t}\sin \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}\textit {f\_1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right ) \cos \left (t \right ) \\
y &= \textit {f\_1} \left (0\right ) \cos \left (t \right )+\left (\int _{0}^{t}\cos \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}\textit {f\_1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right ) \cos \left (t \right )+\left (\int _{0}^{t}\sin \left (\textit {\_z1} \right ) \left (\frac {d}{d \textit {\_z1}}\textit {f\_1} \left (\textit {\_z1} \right )+f_{2} \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right ) \sin \left (t \right )-\textit {f\_1} \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.036 (sec). Leaf size: 216
ode={D[x[t],t]==y[t]+f1[t],D[y[t],t]==-x[t]+f2[t]};
ic={x[0]==0,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(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 ) \\
y(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: 0.385 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-f_1(t) - y(t) + Derivative(x(t), t),0),Eq(-f__2(t) + x(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + \sin {\left (t \right )} \int \left (f_{1}{\left (t \right )} \sin {\left (t \right )} + f^{2}{\left (t \right )} \cos {\left (t \right )}\right )\, dt + \cos {\left (t \right )} \int \left (f_{1}{\left (t \right )} \cos {\left (t \right )} - f^{2}{\left (t \right )} \sin {\left (t \right )}\right )\, dt, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} - \sin {\left (t \right )} \int \left (f_{1}{\left (t \right )} \cos {\left (t \right )} - f^{2}{\left (t \right )} \sin {\left (t \right )}\right )\, dt + \cos {\left (t \right )} \int \left (f_{1}{\left (t \right )} \sin {\left (t \right )} + f^{2}{\left (t \right )} \cos {\left (t \right )}\right )\, dt\right ]
\]