8.13.6 problem problem 12
Internal
problem
ID
[927]
Book
:
Differential
equations
and
linear
algebra,
3rd
ed.,
Edwards
and
Penney
Section
:
Section
7.2,
Matrices
and
Linear
systems.
Page
417
Problem
number
:
problem
12
Date
solved
:
Wednesday, February 05, 2025 at 04:48:31 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right )+1\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{3} \left (t \right )+x_{4} \left (t \right )+t\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+x_{4} \left (t \right )+t^{2}\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+t^{3} \end{align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 272
dsolve([diff(x__1(t),t)=x__2(t)+x__3(t)+1,diff(x__2(t),t)=x__3(t)+x__4(t)+t,diff(x__3(t),t)=x__1(t)+x__4(t)+t^2,diff(x__4(t),t)=x__1(t)+x__2(t)+t^3],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= \frac {t^{2}}{16}-\frac {7 t^{3}}{24}-\frac {t^{4}}{16}+\frac {{\mathrm e}^{2 t} c_1}{2}-\frac {11 t}{16}+c_4 +\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_2}{2}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_3}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_2}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_3}{2} \\
x_{2} \left (t \right ) &= \frac {t^{4}}{16}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_2}{2}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_3}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_2}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_3}{2}-\frac {11 t^{3}}{24}+\frac {{\mathrm e}^{2 t} c_1}{2}+\frac {t^{2}}{16}-c_4 -\frac {3 t}{16}-\frac {19}{16} \\
x_{3} \left (t \right ) &= -\frac {t^{4}}{16}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_2}{2}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_3}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_2}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_3}{2}+\frac {5 t^{3}}{24}+\frac {{\mathrm e}^{2 t} c_1}{2}-\frac {15 t^{2}}{16}+c_4 +\frac {5 t}{16}-\frac {1}{2} \\
x_{4} \left (t \right ) &= \frac {t^{4}}{16}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_2}{2}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_3}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_2}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_3}{2}+\frac {t^{3}}{24}+\frac {{\mathrm e}^{2 t} c_1}{2}-\frac {7 t^{2}}{16}-c_4 -\frac {19 t}{16}+\frac {5}{16} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.299 (sec). Leaf size: 442
DSolve[{D[ x1[t],t]==x2[t]+x3[t]+1,D[ x2[t],t]==x3[t]+x4[t]+t,D[ x3[t],t]==x1[t]+x4[t]+t^2,D[ x4[t],t]==x1[t]+x2[t]+t^3},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (-6 t^4-28 t^3+6 t^2-66 t+3 \left (8 c_1 \left (e^{2 t}+1\right )+8 c_2 \left (e^{2 t}-1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-3+8 c_3-8 c_4\right )\right )+48 (c_1-c_3) \cos (t)+48 (c_2-c_4) \sin (t)\right ) \\
\text {x2}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (6 t^4-44 t^3+6 t^2-18 t+3 \left (8 c_1 \left (e^{2 t}-1\right )+8 c_2 \left (e^{2 t}+1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-35-8 c_3+8 c_4\right )\right )+48 (c_2-c_4) \cos (t)-48 (c_1-c_3) \sin (t)\right ) \\
\text {x3}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (-6 t^4+20 t^3-90 t^2+30 t+3 \left (8 c_1 \left (e^{2 t}+1\right )+8 c_2 \left (e^{2 t}-1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-19+8 c_3-8 c_4\right )\right )-48 (c_1-c_3) \cos (t)-48 (c_2-c_4) \sin (t)\right ) \\
\text {x4}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (6 t^4+4 t^3-42 t^2-114 t+3 \left (8 c_1 \left (e^{2 t}-1\right )+8 c_2 \left (e^{2 t}+1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}+13-8 c_3+8 c_4\right )\right )-48 (c_2-c_4) \cos (t)+48 (c_1-c_3) \sin (t)\right ) \\
\end{align*}