76.6.3 problem 3
Internal
problem
ID
[17379]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
3.
Systems
of
two
first
order
equations.
Section
3.2
(Two
first
order
linear
differential
equations).
Problems
at
page
142
Problem
number
:
3
Date
solved
:
Friday, March 14, 2025 at 04:49:31 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 t x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.562 (sec). Leaf size: 173
ode:=[diff(x(t),t) = -2*t*x(t)+y(t), diff(y(t),t) = 3*x(t)-y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} \operatorname {KummerU}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right )+c_{1} \operatorname {KummerM}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right )\right ) \\
y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (16 \operatorname {KummerU}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right ) c_{2} t^{2}+16 \operatorname {KummerM}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right ) t^{2} c_{1} -16 \operatorname {KummerU}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right ) c_{2} t -16 \operatorname {KummerM}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right ) t c_{1} +8 c_{2} \operatorname {KummerU}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right )-4 \operatorname {KummerM}\left (\frac {3}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right ) c_{1} -c_{2} \operatorname {KummerU}\left (\frac {3}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right )+8 c_{1} \operatorname {KummerM}\left (-\frac {1}{4}, \frac {1}{2}, -t^{2}+t -\frac {1}{4}\right )\right )}{8 t -4} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 143
ode={D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==3*x[t]-y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (3+\sqrt {13}\right ) t} \left (c_1 \left (-\left (\sqrt {13}-13\right ) e^{\sqrt {13} t}+13+\sqrt {13}\right )+2 \sqrt {13} c_2 \left (e^{\sqrt {13} t}-1\right )\right ) \\
y(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (3+\sqrt {13}\right ) t} \left (6 \sqrt {13} c_1 \left (e^{\sqrt {13} t}-1\right )+c_2 \left (\left (13+\sqrt {13}\right ) e^{\sqrt {13} t}+13-\sqrt {13}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.089 (sec). Leaf size: 87
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(2*t*x(t) - y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} x_{0}{\left (t \right )} + C_{2} x_{0}{\left (t \right )} \int \frac {\left (e^{\int \left (-1\right )\, dt}\right ) e^{\int \left (- 2 t\right )\, dt}}{x_{0}^{2}{\left (t \right )}}\, dt, \ y{\left (t \right )} = C_{1} y_{0}{\left (t \right )} + C_{2} \left (y_{0}{\left (t \right )} \int \frac {\left (e^{\int \left (-1\right )\, dt}\right ) e^{\int \left (- 2 t\right )\, dt}}{x_{0}^{2}{\left (t \right )}}\, dt + \frac {\left (e^{\int \left (-1\right )\, dt}\right ) e^{\int \left (- 2 t\right )\, dt}}{x_{0}{\left (t \right )}}\right )\right ]
\]