10.18.2 problem 2
Internal
problem
ID
[1429]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Chapter
7.9,
Nonhomogeneous
Linear
Systems.
page
447
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 04:57:11 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+\sqrt {3}\, x_{2} \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=\sqrt {3}\, x_{1} \left (t \right )-x_{2} \left (t \right )+\sqrt {3}\, {\mathrm e}^{-t} \end{align*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 70
dsolve([diff(x__1(t),t)=1*x__1(t)+sqrt(3)*x__2(t)+exp(t),diff(x__2(t),t)=sqrt(3)*x__1(t)-1*x__2(t)+sqrt(3)*exp(-t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= \sinh \left (2 t \right ) c_2 +\cosh \left (2 t \right ) c_1 -\frac {5 \cosh \left (t \right )}{3}+\frac {\sinh \left (t \right )}{3} \\
x_{2} \left (t \right ) &= -\frac {\sqrt {3}\, \left (\cosh \left (2 t \right ) c_1 -2 \cosh \left (2 t \right ) c_2 -2 \sinh \left (2 t \right ) c_1 +\sinh \left (2 t \right ) c_2 +{\mathrm e}^{t}+2 \sinh \left (t \right )-2 \cosh \left (t \right )\right )}{3} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.705 (sec). Leaf size: 313
DSolve[{D[ x1[t],t]==1*x1[t]+Sqrt[4]*x2[t]+Exp[t],D[ x2[t],t]==Sqrt[3]*x1[t]-1*x2[t]+Sqrt[3]*Exp[-t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {1}{6} \left (-6 e^{-t}-\frac {2 \left (6+\sqrt {3}\right ) e^t}{1+2 \sqrt {3}}+\frac {\left (3 \left (\sqrt {1+2 \sqrt {3}}-1\right ) c_1-6 c_2\right ) e^{-\sqrt {1+2 \sqrt {3}} t}}{\sqrt {1+2 \sqrt {3}}}+\frac {3 \left (\left (1+\sqrt {1+2 \sqrt {3}}\right ) c_1+2 c_2\right ) e^{\sqrt {1+2 \sqrt {3}} t}}{\sqrt {1+2 \sqrt {3}}}\right ) \\
\text {x2}(t)\to \frac {1}{4} \left (4 e^{-t}-2 e^t+\frac {2 \left (\left (6+\sqrt {3}\right ) c_1+\left (1+2 \sqrt {3}\right ) \left (\sqrt {1+2 \sqrt {3}}-1\right ) c_2\right ) e^{\sqrt {1+2 \sqrt {3}} t}}{\left (1+2 \sqrt {3}\right )^{3/2}}+\frac {\left (2 \left (1+2 \sqrt {3}\right ) \left (1+\sqrt {1+2 \sqrt {3}}\right ) c_2-2 \left (6+\sqrt {3}\right ) c_1\right ) e^{-\sqrt {1+2 \sqrt {3}} t}}{\left (1+2 \sqrt {3}\right )^{3/2}}\right ) \\
\end{align*}