28.4.22 problem 7.22
Internal
problem
ID
[4554]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
7.
Systems
of
linear
differential
equations.
Problems
at
page
351
Problem
number
:
7.22
Date
solved
:
Monday, January 27, 2025 at 09:23:26 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )-4 x \left (t \right )+3 y&=\sin \left (t \right )\\ 2 x \left (t \right )+y^{\prime }-y&=2 \cos \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = x_{0}\\ y \left (0\right ) = y_{0} \end{align*}
✓ Solution by Maple
Time used: 0.806 (sec). Leaf size: 217
dsolve([diff(x(t),t)-4*x(t)+3*y(t) = sin(t), 2*x(t)+diff(y(t),t)-y(t) = 2*cos(t), x(0) = x__0, y(0) = y__0], singsol=all)
\begin{align*}
x &= -\frac {{\mathrm e}^{\frac {\left (5+\sqrt {33}\right ) t}{2}} \sqrt {33}\, \left (-17 \sqrt {33}\, x_{0} +3+5 \sqrt {33}-51 x_{0} +102 y_{0} \right )}{1122}+\frac {{\mathrm e}^{-\frac {\left (-5+\sqrt {33}\right ) t}{2}} \sqrt {33}\, \left (17 \sqrt {33}\, x_{0} -5 \sqrt {33}-51 x_{0} +102 y_{0} +3\right )}{1122}+\frac {5 \cos \left (t \right )}{17}+\frac {14 \sin \left (t \right )}{17} \\
y &= \frac {{\mathrm e}^{\frac {\left (5+\sqrt {33}\right ) t}{2}} \left (-17 \sqrt {33}\, x_{0} +3+5 \sqrt {33}-51 x_{0} +102 y_{0} \right )}{204}+\frac {{\mathrm e}^{-\frac {\left (-5+\sqrt {33}\right ) t}{2}} \left (17 \sqrt {33}\, x_{0} -5 \sqrt {33}-51 x_{0} +102 y_{0} +3\right )}{204}-\frac {{\mathrm e}^{\frac {\left (5+\sqrt {33}\right ) t}{2}} \sqrt {33}\, \left (-17 \sqrt {33}\, x_{0} +3+5 \sqrt {33}-51 x_{0} +102 y_{0} \right )}{2244}+\frac {{\mathrm e}^{-\frac {\left (-5+\sqrt {33}\right ) t}{2}} \sqrt {33}\, \left (17 \sqrt {33}\, x_{0} -5 \sqrt {33}-51 x_{0} +102 y_{0} +3\right )}{2244}+\frac {2 \cos \left (t \right )}{17}+\frac {26 \sin \left (t \right )}{17} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.306 (sec). Leaf size: 191
DSolve[{D[x[t],t]-4*x[t]+3*y[t]==Sin[t],2*x[t]+D[y[t],t]-y[t]==2*Cos[t]},{x[0]==x0,y[0]==y0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{374} \left (e^{-\frac {1}{2} \left (\sqrt {33}-5\right ) t} \left (e^{\sqrt {33} t} \left (17 \left (11+\sqrt {33}\right ) \text {x0}-34 \sqrt {33} \text {y0}-\sqrt {33}-55\right )-17 \left (\sqrt {33}-11\right ) \text {x0}+34 \sqrt {33} \text {y0}+\sqrt {33}-55\right )+308 \sin (t)+110 \cos (t)\right ) \\
y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {33}-5\right ) t} \left (e^{\sqrt {33} t} \left (-68 \sqrt {33} \text {x0}-51 \left (\sqrt {33}-11\right ) \text {y0}+26 \sqrt {33}-66\right )+68 \sqrt {33} \text {x0}+51 \sqrt {33} \text {y0}+561 \text {y0}-26 \sqrt {33}-66\right )+1716 \sin (t)+132 \cos (t)}{1122} \\
\end{align*}