76.6.7 problem 7
Internal
problem
ID
[17462]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
3.
Systems
of
two
first
order
equations.
Section
3.2
(Two
first
order
linear
differential
equations).
Problems
at
page
142
Problem
number
:
7
Date
solved
:
Tuesday, January 28, 2025 at 08:27:41 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )+4\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+\sin \left (t \right ) y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 9.319 (sec). Leaf size: 4118
dsolve([diff(x(t),t)=x(t)+y(t)+4,diff(y(t),t)=-2*x(t)+sin(t)*y(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= -16 i \left (\int \frac {\operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \left (-\sin \left (2 t \right )+2 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}+\frac {1}{2}+\frac {i \sin \left (t \right )}{2}+\frac {\cos \left (t \right )}{2}}}{8 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \cos \left (t \right )-4 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \cos \left (2 t \right )-4 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )-8 \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunDPrime}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )+8 \operatorname {HeunDPrime}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )}d t \right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {\cos \left (t \right )}{2}-\frac {1}{2}+\frac {t}{2}-\frac {i \sin \left (t \right )}{2}}+16 i \left (\int \frac {\operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \left (-\sin \left (2 t \right )+2 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}+\frac {1}{2}-\frac {i \sin \left (t \right )}{2}+\frac {\cos \left (t \right )}{2}}}{8 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \cos \left (t \right )-4 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \cos \left (2 t \right )-4 \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )-8 \operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunDPrime}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )+8 \operatorname {HeunDPrime}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right )}d t \right ) \operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{\frac {t}{2}-\frac {1}{2}+\frac {i \sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}}+\operatorname {HeunD}\left (2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {\cos \left (t \right )}{2}-\frac {1}{2}+\frac {t}{2}-\frac {i \sin \left (t \right )}{2}} c_{2} +\operatorname {HeunD}\left (-2, -8, -4 i, 4, i \cot \left (\frac {t}{2}\right )\right ) {\mathrm e}^{\frac {t}{2}-\frac {1}{2}+\frac {i \sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}} c_{1} \\
\text {Expression too large to display} \\
\end{align*}
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[{D[x[t],t]==x[t]+y[t],D[y[t],t]==-2*x[t]+Sin[t]*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
Not solved