50.4.70 problem 67
Internal
problem
ID
[10245]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
67
Date
solved
:
Tuesday, September 30, 2025 at 07:13:07 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+2 t +1\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+y \left (t \right )+3 t -1 \end{align*}
✓ Maple. Time used: 0.127 (sec). Leaf size: 67
ode:=[diff(x(t),t) = x(t)+2*y(t)+2*t+1, diff(y(t),t) = 5*x(t)+y(t)+3*t-1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_2 +{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_1 -\frac {4 t}{9}+\frac {17}{81} \\
y \left (t \right ) &= \frac {{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_2 \sqrt {10}}{2}-\frac {{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_1 \sqrt {10}}{2}-\frac {7 t}{9}-\frac {67}{81} \\
\end{align*}
✓ Mathematica. Time used: 10.842 (sec). Leaf size: 494
ode={D[x[t],t]==x[t]+2*y[t]+2*t+1,D[y[t],t]==5*x[t]+y[t]+3*t-1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{10} e^{t-\sqrt {10} t} \left (5 \left (e^{2 \sqrt {10} t}+1\right ) \int _1^t\frac {1}{10} e^{-\left (\left (1+\sqrt {10}\right ) K[1]\right )} \left (\left (10+3 \sqrt {10}\right ) K[1]+e^{2 \sqrt {10} K[1]} \left (\left (10-3 \sqrt {10}\right ) K[1]+\sqrt {10}+5\right )-\sqrt {10}+5\right )dK[1]+\sqrt {10} \left (e^{2 \sqrt {10} t}-1\right ) \int _1^t\frac {1}{4} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (3+\sqrt {10}\right ) K[2]-e^{2 \sqrt {10} K[2]} \left (2 \left (-3+\sqrt {10}\right ) K[2]+\sqrt {10}+2\right )+\sqrt {10}-2\right )dK[2]+5 c_1 e^{2 \sqrt {10} t}+\sqrt {10} c_2 e^{2 \sqrt {10} t}+5 c_1-\sqrt {10} c_2\right )\\ y(t)&\to \frac {1}{4} e^{t-\sqrt {10} t} \left (\sqrt {10} \left (e^{2 \sqrt {10} t}-1\right ) \int _1^t\frac {1}{10} e^{-\left (\left (1+\sqrt {10}\right ) K[1]\right )} \left (\left (10+3 \sqrt {10}\right ) K[1]+e^{2 \sqrt {10} K[1]} \left (\left (10-3 \sqrt {10}\right ) K[1]+\sqrt {10}+5\right )-\sqrt {10}+5\right )dK[1]+2 \left (e^{2 \sqrt {10} t}+1\right ) \int _1^t\frac {1}{4} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (3+\sqrt {10}\right ) K[2]-e^{2 \sqrt {10} K[2]} \left (2 \left (-3+\sqrt {10}\right ) K[2]+\sqrt {10}+2\right )+\sqrt {10}-2\right )dK[2]+\sqrt {10} c_1 e^{2 \sqrt {10} t}+2 c_2 e^{2 \sqrt {10} t}-\sqrt {10} c_1+2 c_2\right ) \end{align*}
✓ Sympy. Time used: 0.302 (sec). Leaf size: 82
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*t - x(t) - 2*y(t) + Derivative(x(t), t) - 1,0),Eq(-3*t - 5*x(t) - y(t) + Derivative(y(t), t) + 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {\sqrt {10} C_{1} e^{t \left (1 + \sqrt {10}\right )}}{5} - \frac {\sqrt {10} C_{2} e^{t \left (1 - \sqrt {10}\right )}}{5} - \frac {4 t}{9} + \frac {17}{81}, \ y{\left (t \right )} = C_{1} e^{t \left (1 + \sqrt {10}\right )} + C_{2} e^{t \left (1 - \sqrt {10}\right )} - \frac {7 t}{9} - \frac {67}{81}\right ]
\]