87.20.7 problem 7
Internal
problem
ID
[23692]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
3.
Linear
Systems.
Exercise
at
page
161
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 09:44:18 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=3 x-2 y \left (t \right )+2 t^{2}\\ y^{\prime }\left (t \right )&=5 x+y \left (t \right )-1 \end{align*}
✓ Maple. Time used: 0.074 (sec). Leaf size: 91
ode:=[diff(x(t),t) = 3*x(t)-2*y(t)+2*t^2, diff(y(t),t) = 5*x(t)+y(t)-1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{2 t} \sin \left (3 t \right ) c_2 +{\mathrm e}^{2 t} \cos \left (3 t \right ) c_1 -\frac {2 t^{2}}{13}+\frac {36 t}{169}+\frac {534}{2197} \\
y \left (t \right ) &= \frac {10 t^{2}}{13}+\frac {{\mathrm e}^{2 t} \sin \left (3 t \right ) c_2}{2}-\frac {3 \,{\mathrm e}^{2 t} \cos \left (3 t \right ) c_2}{2}+\frac {{\mathrm e}^{2 t} \cos \left (3 t \right ) c_1}{2}+\frac {3 \,{\mathrm e}^{2 t} \sin \left (3 t \right ) c_1}{2}+\frac {80 t}{169}+\frac {567}{2197} \\
\end{align*}
✓ Mathematica. Time used: 0.248 (sec). Leaf size: 104
ode={D[x[t],t]==3*x[t]-2*y[t]+2*t^2,D[y[t],t]==5*x[t]+y[t]-1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {2 \left (-169 t^2+234 t+267\right )}{2197}+c_1 e^{2 t} \cos (3 t)+\frac {1}{3} (c_1-2 c_2) e^{2 t} \sin (3 t)\\ y(t)&\to \frac {1690 t^2+1040 t+567}{2197}+c_2 e^{2 t} \cos (3 t)+\frac {1}{3} (5 c_1-c_2) e^{2 t} \sin (3 t) \end{align*}
✓ Sympy. Time used: 0.434 (sec). Leaf size: 207
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*t**2 - 3*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - y(t) + Derivative(y(t), t) + 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {2 t^{2} \sin ^{2}{\left (3 t \right )}}{13} - \frac {2 t^{2} \cos ^{2}{\left (3 t \right )}}{13} + \frac {36 t \sin ^{2}{\left (3 t \right )}}{169} + \frac {36 t \cos ^{2}{\left (3 t \right )}}{169} + \left (\frac {C_{1}}{5} - \frac {3 C_{2}}{5}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{2 t} \sin {\left (3 t \right )} + \frac {534 \sin ^{2}{\left (3 t \right )}}{2197} + \frac {534 \cos ^{2}{\left (3 t \right )}}{2197}, \ y{\left (t \right )} = C_{1} e^{2 t} \cos {\left (3 t \right )} - C_{2} e^{2 t} \sin {\left (3 t \right )} + \frac {10 t^{2} \sin ^{2}{\left (3 t \right )}}{13} + \frac {10 t^{2} \cos ^{2}{\left (3 t \right )}}{13} + \frac {80 t \sin ^{2}{\left (3 t \right )}}{169} + \frac {80 t \cos ^{2}{\left (3 t \right )}}{169} + \frac {567 \sin ^{2}{\left (3 t \right )}}{2197} + \frac {567 \cos ^{2}{\left (3 t \right )}}{2197}\right ]
\]