63.23.4 problem 6
Internal
problem
ID
[13244]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
244
Problem
number
:
6
Date
solved
:
Tuesday, January 28, 2025 at 05:12:56 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=3 x+2 y \left (t \right )+3\\ y^{\prime }\left (t \right )&=7 x+5 y \left (t \right )+2 t \end{align*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 90
dsolve([diff(x(t),t)=3*x(t)+2*y(t)+3,diff(y(t),t)=7*x(t)+5*y(t)+2*t],singsol=all)
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2} +{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1} +4 t +17 \\
y &= \frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2} \sqrt {15}}{2}-\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1} \sqrt {15}}{2}+\frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2}}{2}+\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1}}{2}-6 t -25 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.702 (sec). Leaf size: 468
DSolve[{D[x[t],t]==3*x[t]+2*y[t],D[y[t],t]==7*x[t]+5*y[t]+2*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{30} e^{-2 \sqrt {15} t} \left (-2 \sqrt {15} \left (e^{\left (4+\sqrt {15}\right ) t}-e^{\left (4+3 \sqrt {15}\right ) t}\right ) \int _1^t-\frac {1}{15} e^{-\left (\left (4+\sqrt {15}\right ) K[1]\right )} \left (-15-\sqrt {15}+\left (-15+\sqrt {15}\right ) e^{2 \sqrt {15} K[1]}\right ) K[1]dK[1]-6 \sqrt {15} t+22 t+e^{2 \sqrt {15} t} (76 t+604)+2 e^{4 \sqrt {15} t} \left (\left (11+3 \sqrt {15}\right ) t+23 \sqrt {15}+89\right )+\left (\left (15+\sqrt {15}\right ) c_1-2 \sqrt {15} c_2\right ) e^{\left (4+\sqrt {15}\right ) t}+\left (2 \sqrt {15} c_2-\left (\sqrt {15}-15\right ) c_1\right ) e^{\left (4+3 \sqrt {15}\right ) t}-46 \sqrt {15}+178\right ) \\
y(t)\to \frac {1}{30} e^{-\left (\left (\sqrt {15}-4\right ) t\right )} \left (\left (\left (15+\sqrt {15}\right ) e^{2 \sqrt {15} t}+15-\sqrt {15}\right ) \int _1^t-\frac {1}{15} e^{-\left (\left (4+\sqrt {15}\right ) K[1]\right )} \left (-15-\sqrt {15}+\left (-15+\sqrt {15}\right ) e^{2 \sqrt {15} K[1]}\right ) K[1]dK[1]+14 e^{-\left (\left (4+\sqrt {15}\right ) t\right )} \left (e^{2 \sqrt {15} t}-1\right ) \left (\left (\sqrt {15}-4\right ) t+e^{2 \sqrt {15} t} \left (\left (4+\sqrt {15}\right ) t+8 \sqrt {15}+31\right )+8 \sqrt {15}-31\right )+7 \sqrt {15} c_1 \left (e^{2 \sqrt {15} t}-1\right )+c_2 \left (\left (15+\sqrt {15}\right ) e^{2 \sqrt {15} t}+15-\sqrt {15}\right )\right ) \\
\end{align*}