60.9.27 problem 1882
Internal
problem
ID
[11806]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1882
Date
solved
:
Friday, March 14, 2025 at 02:58:22 AM
CAS
classification
:
system_of_ODEs
\begin{align*} 2 \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-3 x \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}
✓ Maple. Time used: 0.095 (sec). Leaf size: 117
ode:=[2*diff(x(t),t)+diff(y(t),t)-3*x(t) = 0, diff(diff(x(t),t),t)+diff(y(t),t)-2*y(t) = exp(2*t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{4}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) \\
y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{8}+c_{1} {\mathrm e}^{t}-\frac {7 c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {7 c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 2.301 (sec). Leaf size: 932
ode={2*D[x[t],t]+D[y[t],t]-3*x[t]==0,D[x[t],{t,2}]+D[y[t],t]-2*y[t]==Exp[2*t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 0.582 (sec). Leaf size: 197
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) + 2*Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-2*y(t) - exp(2*t) + Derivative(x(t), (t, 2)) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{3} e^{t} + \left (\frac {7 C_{1}}{18} - \frac {\sqrt {23} C_{2}}{18}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {23} t}{2} \right )} - \left (\frac {\sqrt {23} C_{1}}{18} + \frac {7 C_{2}}{18}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {23} t}{2} \right )} + \frac {e^{2 t} \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{12} + \frac {e^{2 t} \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{12} + \frac {e^{2 t}}{6}, \ y{\left (t \right )} = - C_{1} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {23} t}{2} \right )} + C_{2} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {23} t}{2} \right )} + C_{3} e^{t} - \frac {7 e^{2 t} \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{24} - \frac {7 e^{2 t} \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{24} + \frac {e^{2 t}}{6}\right ]
\]