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
:
Monday, January 27, 2025 at 06:12:09 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=y+\textit {f\_1} \left (t \right )\\ y^{\prime }&=-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*}
✓ Solution by Maple
Time used: 1.849 (sec). Leaf size: 100
dsolve([diff(x(t),t) = y(t)+f_1(t), diff(y(t),t) = -x(t)+f__2(t), x(0) = 0, y(0) = 0], singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 216
DSolve[{D[x[t],t]==y[t]+f1[t],D[y[t],t]==-x[t]+f2[t]},{x[0]==0,y[0]==0},{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*}