85.83.11 problem 4 (b)
Internal
problem
ID
[23008]
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
:
4
(b)
Date
solved
:
Sunday, October 12, 2025 at 05:55:05 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+3 \frac {d}{d t}y \left (t \right )&=x \left (t \right ) y \left (t \right )\\ 3 \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )&=\sin \left (t \right ) \end{align*}
✓ Maple. Time used: 1.497 (sec). Leaf size: 359
ode:=[diff(x(t),t)+3*diff(y(t),t) = x(t)*y(t), 3*diff(x(t),t)-diff(y(t),t) = sin(t)];
dsolve(ode);
\begin{align*}
\left \{x \left (t \right ) &= \frac {\left (5 i \operatorname {HeunDPrime}\left (\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) \cot \left (\frac {t}{2}\right )^{2} c_2 -\cos \left (t \right ) \operatorname {HeunD}\left (\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) c_2 +5 i \operatorname {HeunDPrime}\left (\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) c_2 +5 \operatorname {HeunD}\left (\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) c_1 c_2 \right ) {\mathrm e}^{-\frac {c_1 t}{2}+\frac {\sin \left (t \right )}{10}}+\left (5 i \operatorname {HeunDPrime}\left (-\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) \cot \left (\frac {t}{2}\right )^{2}+5 i \operatorname {HeunDPrime}\left (-\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right )+5 \operatorname {HeunD}\left (-\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) c_1 \right ) {\mathrm e}^{-\frac {c_1 t}{2}}}{3 c_2 \operatorname {HeunD}\left (\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {c_1 t}{2}+\frac {\sin \left (t \right )}{10}}+3 \operatorname {HeunD}\left (-\frac {i}{5}, -\frac {1}{5} c_1 +c_1^{2}, -\frac {8 i}{25}, -\frac {1}{5} c_1 -c_1^{2}, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {c_1 t}{2}}}\right \} \\
\left \{y \left (t \right ) &= \frac {-3 \sin \left (t \right )+10 \frac {d}{d t}x \left (t \right )}{x \left (t \right )}\right \} \\
\end{align*}
✗ Mathematica
ode={D[x[t],{t,1}]+3*D[y[t],t]==x[t]*y[t],3*D[x[t],{t,1}]-D[y[t],t]==Sin[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t)*y(t) + Derivative(x(t), t) + 3*Derivative(y(t), t),0),Eq(-sin(t) + 3*Derivative(x(t), t) - Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
NotImplementedError :