2.13.3 problem problem 5
Internal
problem
ID
[924]
Book
:
Differential
equations
and
linear
algebra,
3rd
ed.,
Edwards
and
Penney
Section
:
Section
7.2,
Matrices
and
Linear
systems.
Page
417
Problem
number
:
problem
5
Date
solved
:
Tuesday, September 30, 2025 at 04:19:39 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+4 y \left (t \right )+3 \,{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )-y \left (t \right )-t^{2} \end{align*}
✓ Maple. Time used: 0.223 (sec). Leaf size: 111
ode:=[diff(x(t),t) = 2*x(t)+4*y(t)+3*exp(t), diff(y(t),t) = 5*x(t)-y(t)-t^2];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_2 \sqrt {89}}{10}-\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_1 \sqrt {89}}{10}+\frac {3 \,{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_2}{10}+\frac {3 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_1}{10}+\frac {2 t^{2}}{11}-\frac {3 \,{\mathrm e}^{t}}{11}-\frac {2 t}{121}+\frac {23}{1331} \\
y \left (t \right ) &= {\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_2 +{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_1 -\frac {t^{2}}{11}-\frac {15 \,{\mathrm e}^{t}}{22}+\frac {12 t}{121}-\frac {17}{1331} \\
\end{align*}
✓ Mathematica. Time used: 0.185 (sec). Leaf size: 212
ode={D[x[t],t]==2*x[t]+4*y[t]+3*Exp[t],D[y[t],t]==5*x[t]-y[t]-t^2};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {242 t^2-22 t+23}{1331}-\frac {3 e^t}{11}+\frac {1}{178} \left (\left (89-3 \sqrt {89}\right ) c_1-8 \sqrt {89} c_2\right ) e^{-\frac {1}{2} \left (\sqrt {89}-1\right ) t}+\frac {1}{178} \left (\left (89+3 \sqrt {89}\right ) c_1+8 \sqrt {89} c_2\right ) e^{\frac {1}{2} \left (1+\sqrt {89}\right ) t}\\ y(t)&\to \frac {-121 t^2+132 t-17}{1331}-\frac {15 e^t}{22}+\left (\frac {5 c_1}{\sqrt {89}}+\frac {1}{178} \left (89-3 \sqrt {89}\right ) c_2\right ) e^{\frac {1}{2} \left (1+\sqrt {89}\right ) t}+\left (\frac {1}{178} \left (89+3 \sqrt {89}\right ) c_2-\frac {5 c_1}{\sqrt {89}}\right ) e^{-\frac {1}{2} \left (\sqrt {89}-1\right ) t} \end{align*}
✓ Sympy. Time used: 5.719 (sec). Leaf size: 117
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) - 4*y(t) - 3*exp(t) + Derivative(x(t), t),0),Eq(t**2 - 5*x(t) + y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (3 + \sqrt {89}\right ) e^{\frac {t \left (1 + \sqrt {89}\right )}{2}}}{10} + \frac {C_{2} \left (3 - \sqrt {89}\right ) e^{\frac {t \left (1 - \sqrt {89}\right )}{2}}}{10} + \frac {2 t^{2}}{11} - \frac {2 t}{121} - \frac {3 e^{t}}{11} + \frac {23}{1331}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (1 + \sqrt {89}\right )}{2}} + C_{2} e^{\frac {t \left (1 - \sqrt {89}\right )}{2}} - \frac {t^{2}}{11} + \frac {12 t}{121} - \frac {15 e^{t}}{22} - \frac {17}{1331}\right ]
\]