52.9.10 problem 10
Internal
problem
ID
[8388]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
8
SYSTEMS
OF
LINEAR
FIRST-ORDER
DIFFERENTIAL
EQUATIONS.
EXERCISES
8.1.
Page
332
Problem
number
:
10
Date
solved
:
Monday, January 27, 2025 at 03:57:18 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-7 y+4 \sin \left (t \right )+\left (t -4\right ) {\mathrm e}^{4 t}\\ y^{\prime }&=x \left (t \right )+y+8 \sin \left (t \right )+\left (2 t +1\right ) {\mathrm e}^{4 t} \end{align*}
✓ Solution by Maple
Time used: 2.350 (sec). Leaf size: 133
dsolve([diff(x(t),t)=3*x(t)-7*y(t)+4*sin(t)+(t-4)*exp(4*t),diff(y(t),t)=x(t)+y(t)+8*sin(t)+(2*t+1)*exp(4*t)],singsol=all)
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2} +{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1} -\frac {11 \,{\mathrm e}^{4 t} t}{10}-\frac {34 \,{\mathrm e}^{4 t}}{25}-\frac {204 \cos \left (t \right )}{97}-\frac {556 \sin \left (t \right )}{97} \\
y &= \frac {3 \,{\mathrm e}^{4 t} t}{10}+\frac {{\mathrm e}^{2 t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right ) c_{1}}{7}-\frac {{\mathrm e}^{2 t} \sqrt {6}\, \cos \left (\sqrt {6}\, t \right ) c_{2}}{7}-\frac {11 \,{\mathrm e}^{4 t}}{50}+\frac {{\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}}{7}+\frac {{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}}{7}-\frac {8 \cos \left (t \right )}{97}-\frac {212 \sin \left (t \right )}{97} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.715 (sec). Leaf size: 190
DSolve[{D[x[t],t]==3*x[t]-7*y[t]+4*Sin[t]+(t-4)*Exp[4*t],D[y[t],t]==x[t]+y[t]+8*Sin[t]+(2*t+1)*Exp[4*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to -\frac {11}{10} e^{4 t} t-\frac {34 e^{4 t}}{25}-\frac {556 \sin (t)}{97}-\frac {204 \cos (t)}{97}+c_1 e^{2 t} \cos \left (\sqrt {6} t\right )+\frac {c_1 e^{2 t} \sin \left (\sqrt {6} t\right )}{\sqrt {6}}-\frac {7 c_2 e^{2 t} \sin \left (\sqrt {6} t\right )}{\sqrt {6}} \\
y(t)\to \frac {3}{10} e^{4 t} t-\frac {11 e^{4 t}}{50}-\frac {212 \sin (t)}{97}-\frac {8 \cos (t)}{97}+c_2 e^{2 t} \cos \left (\sqrt {6} t\right )+\frac {c_1 e^{2 t} \sin \left (\sqrt {6} t\right )}{\sqrt {6}}-\frac {c_2 e^{2 t} \sin \left (\sqrt {6} t\right )}{\sqrt {6}} \\
\end{align*}