7.24.3 problem 13
Internal
problem
ID
[603]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
5.
Linear
systems
of
differential
equations.
Section
5.3
(Matrices
and
linear
systems).
Problems
at
page
364
Problem
number
:
13
Date
solved
:
Monday, January 27, 2025 at 02:55:11 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=2 x+4 y \left (t \right )+3 \,{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=5 x-y \left (t \right )-t^{2} \end{align*}
✓ Solution by Maple
Time used: 0.105 (sec). Leaf size: 111
dsolve([diff(x(t),t)=2*x(t)+4*y(t)+3*exp(t),diff(y(t),t)=5*x(t)-y(t)-t^2],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 1.008 (sec). Leaf size: 212
DSolve[{D[x[t],t]==2*x[t]+4*y[t]+3*Exp[t],D[y[t],t]==5*x[t]-y[t]-t^2},{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*}