57.2.2 problem 2
Internal
problem
ID
[9056]
Book
:
First
order
enumerated
odes
Section
:
section
2
(system
of
first
order
odes)
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 05:31:42 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 2 \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )&=y \left (t \right )+t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t} \end{align*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 94
dsolve([2*diff(x(t),t)+diff(y(t),t)-x(t)=y(t)+t,diff(x(t),t)+diff(y(t),t)=2*x(t)+3*y(t)+exp(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (2+\sqrt {3}\right ) t} c_{2} +{\mathrm e}^{-\left (-2+\sqrt {3}\right ) t} c_{1} -3 t -11 \\
y \left (t \right ) &= -\frac {{\mathrm e}^{\left (2+\sqrt {3}\right ) t} c_{2} \sqrt {3}}{2}+\frac {{\mathrm e}^{-\left (-2+\sqrt {3}\right ) t} c_{1} \sqrt {3}}{2}-\frac {3 \,{\mathrm e}^{\left (2+\sqrt {3}\right ) t} c_{2}}{2}-\frac {3 \,{\mathrm e}^{-\left (-2+\sqrt {3}\right ) t} c_{1}}{2}-\frac {{\mathrm e}^{t}}{2}+2 t +7 \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.148 (sec). Leaf size: 262
DSolve[{2*D[x[t],t]+D[y[t],t]-x[t]==y[t]+t,D[x[t],t]+D[y[t],t]==2*x[t]+3*y[t]+Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {e^{-\sqrt {3} t} \left (-6 \left (2+\sqrt {3}\right ) e^{\sqrt {3} t} (3 t+11)-\left (3 \left (1+\sqrt {3}\right ) c_1+2 \left (3+2 \sqrt {3}\right ) c_2\right ) e^{2 \left (1+\sqrt {3}\right ) t}+\left (3 \left (5+3 \sqrt {3}\right ) c_1+2 \left (3+2 \sqrt {3}\right ) c_2\right ) e^{2 t}\right )}{6 \left (2+\sqrt {3}\right )} \\
y(t)\to \frac {e^{-\sqrt {3} t} \left (2 \left (2+\sqrt {3}\right ) e^{\sqrt {3} t} (2 t+7)-\left (2+\sqrt {3}\right ) e^{\left (1+\sqrt {3}\right ) t}-\left (\left (3+2 \sqrt {3}\right ) c_1+\left (1+\sqrt {3}\right ) c_2\right ) e^{2 t}+\left (\left (3+2 \sqrt {3}\right ) c_1+\left (5+3 \sqrt {3}\right ) c_2\right ) e^{2 \left (1+\sqrt {3}\right ) t}\right )}{2 \left (2+\sqrt {3}\right )} \\
\end{align*}