64.16.12 problem 12
Internal
problem
ID
[13615]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
7,
Systems
of
linear
differential
equations.
Section
7.1.
Exercises
page
277
Problem
number
:
12
Date
solved
:
Tuesday, January 28, 2025 at 05:53:54 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )+5 y \left (t \right )&=t^{2}\\ \frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )-2 x \left (t \right )+4 y \left (t \right )&=2 t +1 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 115
dsolve([diff(x(t),t)+diff(y(t),t)-x(t)+5*y(t)=t^2,diff(x(t),t)+2*diff(y(t),t)-2*x(t)+4*y(t)=2*t+1],singsol=all)
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2} +{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1} +\frac {2 t^{2}}{3}-\frac {7 t}{9}-\frac {41}{27} \\
y \left (t \right ) &= \frac {t^{2}}{3}-\frac {{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2}}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2}}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1}}{12}+\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1}}{12}-\frac {5 t}{9}-\frac {1}{27} \\
\end{align*}
✓ Solution by Mathematica
Time used: 6.778 (sec). Leaf size: 540
DSolve[{D[x[t],t]+D[y[t],t]-x[t]+5*y[t]==t^2,D[x[t],t]+2*D[y[t],t]-2*x[t]+4*y[t]==2*t+1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to -\frac {1}{23} e^{t/2} \left (\left (\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{23} e^{-\frac {K[1]}{2}} \left (23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right ) \left (2 K[1]^2-2 K[1]-1\right )+\sqrt {23} \left (-10 K[1]^2+22 K[1]+11\right ) \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+12 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right ) \int _1^t\frac {1}{23} e^{-\frac {K[2]}{2}} \left (\sqrt {23} \left (-3 K[2]^2+2 K[2]+1\right ) \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )-23 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right ) \left (K[2]^2-2 K[2]-1\right )\right )dK[2]-23 c_1 \cos \left (\frac {\sqrt {23} t}{2}\right )+\sqrt {23} c_1 \sin \left (\frac {\sqrt {23} t}{2}\right )+12 \sqrt {23} c_2 \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\
y(t)\to \frac {1}{23} e^{t/2} \left (2 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right ) \int _1^t\frac {1}{23} e^{-\frac {K[1]}{2}} \left (23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right ) \left (2 K[1]^2-2 K[1]-1\right )+\sqrt {23} \left (-10 K[1]^2+22 K[1]+11\right ) \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+\left (\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{23} e^{-\frac {K[2]}{2}} \left (\sqrt {23} \left (-3 K[2]^2+2 K[2]+1\right ) \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )-23 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right ) \left (K[2]^2-2 K[2]-1\right )\right )dK[2]+23 c_2 \cos \left (\frac {\sqrt {23} t}{2}\right )+2 \sqrt {23} c_1 \sin \left (\frac {\sqrt {23} t}{2}\right )+\sqrt {23} c_2 \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\
\end{align*}