63.19.7 problem 3(a)
Internal
problem
ID
[13221]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
202
Problem
number
:
3(a)
Date
solved
:
Tuesday, January 28, 2025 at 05:12:37 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=2 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x-14 \end{align*}
✓ Solution by Maple
Time used: 0.065 (sec). Leaf size: 76
dsolve([diff(x(t),t)=2*x(t)+3*y(t),diff(y(t),t)=-x(t)-14],singsol=all)
\begin{align*}
x \left (t \right ) &= -14+{\mathrm e}^{t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) c_{1} -\sqrt {2}\, \cos \left (\sqrt {2}\, t \right ) c_{2} -\sin \left (\sqrt {2}\, t \right ) c_{2} -\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\
y &= \frac {28}{3}+{\mathrm e}^{t} \left (\sin \left (\sqrt {2}\, t \right ) c_{2} +\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.167 (sec). Leaf size: 336
DSolve[{D[x[t],t]==2*x[t]+3*y[t],D[y[t],t]==-x[t]-14},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{2} e^t \left (\left (\sqrt {2} \sin \left (\sqrt {2} t\right )+2 \cos \left (\sqrt {2} t\right )\right ) \int _1^t21 \sqrt {2} e^{-K[1]} \sin \left (\sqrt {2} K[1]\right )dK[1]+3 \sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^t-7 e^{-K[2]} \left (2 \cos \left (\sqrt {2} K[2]\right )+\sqrt {2} \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+2 c_1 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )+3 \sqrt {2} c_2 \sin \left (\sqrt {2} t\right )\right ) \\
y(t)\to -\frac {1}{2} e^t \left (\sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^t21 \sqrt {2} e^{-K[1]} \sin \left (\sqrt {2} K[1]\right )dK[1]+\left (\sqrt {2} \sin \left (\sqrt {2} t\right )-2 \cos \left (\sqrt {2} t\right )\right ) \int _1^t-7 e^{-K[2]} \left (2 \cos \left (\sqrt {2} K[2]\right )+\sqrt {2} \sin \left (\sqrt {2} K[2]\right )\right )dK[2]-2 c_2 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} t\right )\right ) \\
\end{align*}