86.12.5 problem 6
Internal
problem
ID
[23212]
Book
:
An
introduction
to
Differential
Equations.
By
Howard
Frederick
Cleaves.
1969.
Oliver
and
Boyd
publisher.
ISBN
0050015044
Section
:
Chapter
10.
The
Laplace
transform.
Exercise
10c
at
page
156
Problem
number
:
6
Date
solved
:
Thursday, October 02, 2025 at 09:24:25 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 4 \frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+3 x \left (t \right )&=E \sin \left (t \right )\\ 4 x \left (t \right )+2 \frac {d}{d t}x \left (t \right )+3 y \left (t \right )&=0 \end{align*}
With initial conditions
\begin{align*}
x \left (0\right )&=0 \\
y \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.277 (sec). Leaf size: 149
ode:=[4*diff(x(t),t)+2*diff(y(t),t)+3*x(t) = E*sin(t), 4*x(t)+2*diff(x(t),t)+3*y(t) = 0];
ic:=[x(0) = 0, y(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (1+\sqrt {10}\right ) t}{2}} \left (11 \sqrt {10}+10\right ) E \sqrt {10}}{1850}+\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {10}\right ) t}{2}} \left (-10+11 \sqrt {10}\right ) E \sqrt {10}}{1850}+\frac {39 E \sin \left (t \right )}{185}-\frac {12 E \cos \left (t \right )}{185}-\frac {{\mathrm e}^{\frac {\left (1+\sqrt {10}\right ) t}{2}} \left (11 \sqrt {10}+10\right ) E}{370}+\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {10}\right ) t}{2}} \left (-10+11 \sqrt {10}\right ) E}{370} \\
y \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (1+\sqrt {10}\right ) t}{2}} \left (11 \sqrt {10}+10\right ) E}{370}-\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {10}\right ) t}{2}} \left (-10+11 \sqrt {10}\right ) E}{370}-\frac {12 E \sin \left (t \right )}{37}-\frac {2 E \cos \left (t \right )}{37} \\
\end{align*}
✓ Mathematica. Time used: 0.593 (sec). Leaf size: 122
ode={4*D[x[t],{t,1}]+2*D[y[t],t]+3*x[t]==e*Sin[t],4*x[t]+2*D[x[t],{t,1}]+3*y[t]==0};
ic={x[0]==0,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {3}{370} e \left (e^{-\frac {1}{2} \left (\sqrt {10}-1\right ) t} \left (\left (4-3 \sqrt {10}\right ) e^{\sqrt {10} t}+4+3 \sqrt {10}\right )+26 \sin (t)-8 \cos (t)\right )\\ y(t)&\to \frac {1}{370} e \left (e^{-\frac {1}{2} \left (\sqrt {10}-1\right ) t} \left (\left (10+11 \sqrt {10}\right ) e^{\sqrt {10} t}+10-11 \sqrt {10}\right )-120 \sin (t)-20 \cos (t)\right ) \end{align*}
✓ Sympy. Time used: 1.379 (sec). Leaf size: 502
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-e*sin(t) + 3*x(t) + 4*Derivative(x(t), t) + 2*Derivative(y(t), t),0),Eq(4*x(t) + 3*y(t) + 2*Derivative(x(t), t),0)]
ics = {x(0): 0, y(0): 0}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {3 e \left (4 + 3 \sqrt {10}\right ) e^{\frac {t \left (1 - \sqrt {10}\right )}{2}}}{370} + \frac {3 e \left (4 - 3 \sqrt {10}\right ) e^{\frac {t \left (1 + \sqrt {10}\right )}{2}}}{370} + \frac {3 e \left (5 + 4 \sqrt {10}\right ) e^{\frac {t}{2}} \sin {\left (t \right )}}{10 \left (10 e^{\frac {t}{2}} + 11 \sqrt {10} e^{\frac {t}{2}}\right )} - \frac {3 e \left (\sqrt {10} + 5\right ) e^{\frac {t}{2}} \cos {\left (t \right )}}{5 \left (10 e^{\frac {t}{2}} + 11 \sqrt {10} e^{\frac {t}{2}}\right )} + \frac {3 e \left (\sqrt {10} + 10\right ) e^{\frac {t \left (1 + \sqrt {10}\right )}{2}} \sin {\left (t \right )}}{20 \left (2 \sqrt {10} e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}} + 15 e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}}\right )} + \frac {3 \sqrt {10} e e^{\frac {t \left (1 + \sqrt {10}\right )}{2}} \cos {\left (t \right )}}{10 \left (2 \sqrt {10} e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}} + 15 e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}}\right )}, \ y{\left (t \right )} = \frac {e \left (10 - 11 \sqrt {10}\right ) e^{\frac {t \left (1 - \sqrt {10}\right )}{2}}}{370} + \frac {e \left (10 + 11 \sqrt {10}\right ) e^{\frac {t \left (1 + \sqrt {10}\right )}{2}}}{370} + \frac {3 e \left (1 - \sqrt {10}\right ) e^{\frac {t}{2}} \sin {\left (t \right )}}{2 \left (10 e^{\frac {t}{2}} + 11 \sqrt {10} e^{\frac {t}{2}}\right )} + \frac {3 e e^{\frac {t}{2}} \cos {\left (t \right )}}{10 e^{\frac {t}{2}} + 11 \sqrt {10} e^{\frac {t}{2}}} - \frac {3 e \left (\sqrt {10} + 4\right ) e^{\frac {t \left (1 + \sqrt {10}\right )}{2}} \sin {\left (t \right )}}{4 \left (2 \sqrt {10} e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}} + 15 e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}}\right )} - \frac {e \left (2 + \sqrt {10}\right ) e^{\frac {t \left (1 + \sqrt {10}\right )}{2}} \cos {\left (t \right )}}{2 \left (2 \sqrt {10} e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}} + 15 e^{\frac {t}{2}} e^{\frac {\sqrt {10} t}{2}}\right )}\right ]
\]