66.3.2 problem Problem 3
Internal
problem
ID
[13945]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
3,
SYSTEMS
OF
DIFFERENTIAL
EQUATIONS.
Problems
page
209
Problem
number
:
Problem
3
Date
solved
:
Tuesday, January 28, 2025 at 06:09:20 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+y \left (t \right )&={\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )-3 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 101
dsolve([diff(x(t),t)+5*x(t)+y(t)=exp(t),diff(y(t),t)-x(t)-3*y(t)=exp(2*t)],singsol=all)
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_{2} +{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} +\frac {{\mathrm e}^{2 t}}{6}+\frac {2 \,{\mathrm e}^{t}}{11} \\
y \left (t \right ) &= -{\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_{2} \sqrt {15}+{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} \sqrt {15}-4 \,{\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_{2} -4 \,{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} -\frac {{\mathrm e}^{t}}{11}-\frac {7 \,{\mathrm e}^{2 t}}{6} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.962 (sec). Leaf size: 206
DSolve[{D[x[t],t]+5*x[t]+y[t]==Exp[t],D[y[t],t]-x[t]-3*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (60 e^{\left (2+\sqrt {15}\right ) t}+55 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\left (4 \sqrt {15}-15\right ) c_1+\sqrt {15} c_2\right ) e^{2 \sqrt {15} t}+11 \left (\left (15+4 \sqrt {15}\right ) c_1+\sqrt {15} c_2\right )\right ) \\
y(t)\to -\frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (30 e^{\left (2+\sqrt {15}\right ) t}+385 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\sqrt {15} c_1+\left (15+4 \sqrt {15}\right ) c_2\right ) e^{2 \sqrt {15} t}+11 \left (\sqrt {15} c_1+\left (4 \sqrt {15}-15\right ) c_2\right )\right ) \\
\end{align*}