85.83.7 problem 1 (g)
Internal
problem
ID
[23004]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
10.
Systems
of
differential
equations
and
their
applications.
A
Exercises
at
page
444
Problem
number
:
1
(g)
Date
solved
:
Thursday, October 02, 2025 at 09:17:27 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 y \left (t \right )&=1\\ 2 x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=t \end{align*}
✓ Maple. Time used: 0.058 (sec). Leaf size: 43
ode:=[diff(x(t),t)+x(t)+2*y(t) = 1, 2*x(t)+diff(y(t),t)-2*y(t) = t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-2 t} c_2 +{\mathrm e}^{3 t} c_1 +\frac {t}{3}+\frac {5}{18} \\
y \left (t \right ) &= \frac {{\mathrm e}^{-2 t} c_2}{2}-2 \,{\mathrm e}^{3 t} c_1 +\frac {7}{36}-\frac {t}{6} \\
\end{align*}
✓ Mathematica. Time used: 1.359 (sec). Leaf size: 133
ode={D[x[t],{t,1}]+x[t]+2*y[t]==1,2*x[t]-D[y[t],{t,1}]-2*y[t]==t};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{6} (2 t+1)+c_1 e^{-3 t/2} \cos \left (\frac {\sqrt {15} t}{2}\right )+\frac {(c_1-4 c_2) e^{-3 t/2} \sin \left (\frac {\sqrt {15} t}{2}\right )}{\sqrt {15}}\\ y(t)&\to -\frac {t}{6}+c_2 e^{-3 t/2} \cos \left (\frac {\sqrt {15} t}{2}\right )+\frac {(4 c_1-c_2) e^{-3 t/2} \sin \left (\frac {\sqrt {15} t}{2}\right )}{\sqrt {15}}+\frac {1}{4} \end{align*}
✓ Sympy. Time used: 0.494 (sec). Leaf size: 214
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(x(t) + 2*y(t) + Derivative(x(t), t) - 1,0),Eq(-t + 2*x(t) - 2*y(t) - Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {t \sin ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{3} + \frac {t \cos ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{3} + \left (\frac {C_{1}}{4} - \frac {\sqrt {15} C_{2}}{4}\right ) e^{- \frac {3 t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} - \left (\frac {\sqrt {15} C_{1}}{4} + \frac {C_{2}}{4}\right ) e^{- \frac {3 t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )} + \frac {\sin ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{6} + \frac {\cos ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{6}, \ y{\left (t \right )} = C_{1} e^{- \frac {3 t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} - C_{2} e^{- \frac {3 t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )} - \frac {t \sin ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{6} - \frac {t \cos ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{6} + \frac {\sin ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{4} + \frac {\cos ^{2}{\left (\frac {\sqrt {15} t}{2} \right )}}{4}\right ]
\]