90.33.5 problem 5
Internal
problem
ID
[25479]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
9.
Linear
Systems
of
Differential
Equations.
Exercises
at
page
645
Problem
number
:
5
Date
solved
:
Friday, October 03, 2025 at 12:01:58 AM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (t \right )&=y_{1} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+y_{4} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{4} \left (t \right )\\ y_{4}^{\prime }\left (t \right )&=y_{2} \left (t \right )+2 y_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.107 (sec). Leaf size: 105
ode:=[diff(y__1(t),t) = y__1(t), diff(y__2(t),t) = 2*y__1(t)+y__4(t), diff(y__3(t),t) = y__4(t), diff(y__4(t),t) = y__2(t)+2*y__3(t)];
dsolve(ode);
\begin{align*}
y_{1} \left (t \right ) &= c_4 \,{\mathrm e}^{t} \\
y_{2} \left (t \right ) &= -\frac {\sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t} c_2}{3}+\frac {\sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t} c_3}{3}+c_4 \,{\mathrm e}^{t}-2 c_1 \\
y_{3} \left (t \right ) &= \frac {\sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t} c_3}{3}-\frac {\sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t} c_2}{3}-c_4 \,{\mathrm e}^{t}+c_1 \\
y_{4} \left (t \right ) &= {\mathrm e}^{-\sqrt {3}\, t} c_2 +{\mathrm e}^{\sqrt {3}\, t} c_3 -c_4 \,{\mathrm e}^{t} \\
\end{align*}
✓ Mathematica. Time used: 0.014 (sec). Leaf size: 429
ode={D[y1[t],t]==y1[t], D[y2[t],t]==2*y1[t]+y4[t],D[y3[t],t]==y4[t],D[y4[t],t]==y2[t]+2*y3[t]};
ic={};
DSolve[{ode,ic},{y1[t],y2[t],y3[t],y4[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {y1}(t)&\to c_1 e^t\\ \text {y2}(t)&\to \frac {1}{6} e^{-\sqrt {3} t} \left (c_1 \left (-8 e^{\sqrt {3} t}+\left (1+\sqrt {3}\right ) e^{2 \sqrt {3} t}+6 e^{\sqrt {3} t+t}+1-\sqrt {3}\right )+c_2 \left (4 e^{\sqrt {3} t}+e^{2 \sqrt {3} t}+1\right )+\left (e^{\sqrt {3} t}-1\right ) \left (2 c_3 \left (e^{\sqrt {3} t}-1\right )+\sqrt {3} c_4 \left (e^{\sqrt {3} t}+1\right )\right )\right )\\ \text {y3}(t)&\to \frac {1}{6} e^{-\sqrt {3} t} \left (c_2 \left (e^{\sqrt {3} t}-1\right )^2+c_1 \left (4 e^{\sqrt {3} t}+\left (1+\sqrt {3}\right ) e^{2 \sqrt {3} t}-6 e^{\sqrt {3} t+t}+1-\sqrt {3}\right )+2 c_3 e^{\sqrt {3} t}+2 c_3 e^{2 \sqrt {3} t}+\sqrt {3} c_4 e^{2 \sqrt {3} t}+2 c_3-\sqrt {3} c_4\right )\\ \text {y4}(t)&\to \frac {1}{6} e^{-\sqrt {3} t} \left (c_1 \left (\left (3+\sqrt {3}\right ) e^{2 \sqrt {3} t}-6 e^{\sqrt {3} t+t}+3-\sqrt {3}\right )+\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )+2 \sqrt {3} c_3 e^{2 \sqrt {3} t}+3 c_4 e^{2 \sqrt {3} t}-2 \sqrt {3} c_3+3 c_4\right ) \end{align*}
✓ Sympy. Time used: 0.173 (sec). Leaf size: 122
from sympy import *
t = symbols("t")
y1 = Function("y1")
y2 = Function("y2")
y3 = Function("y3")
y4 = Function("y4")
ode=[Eq(-y1(t) + Derivative(y1(t), t),0),Eq(-2*y1(t) - y4(t) + Derivative(y2(t), t),0),Eq(-y4(t) + Derivative(y3(t), t),0),Eq(-y2(t) - 2*y3(t) + Derivative(y4(t), t),0)]
ics = {}
dsolve(ode,func=[y1(t),y2(t),y3(t),y4(t)],ics=ics)
\[
\left [ y_{1}{\left (t \right )} = - C_{1} e^{t}, \ y_{2}{\left (t \right )} = - C_{1} e^{t} - 2 C_{2} - \frac {\sqrt {3} C_{3} e^{- \sqrt {3} t}}{3} + \frac {\sqrt {3} C_{4} e^{\sqrt {3} t}}{3}, \ y_{3}{\left (t \right )} = C_{1} e^{t} + C_{2} - \frac {\sqrt {3} C_{3} e^{- \sqrt {3} t}}{3} + \frac {\sqrt {3} C_{4} e^{\sqrt {3} t}}{3}, \ y_{4}{\left (t \right )} = C_{1} e^{t} + C_{3} e^{- \sqrt {3} t} + C_{4} e^{\sqrt {3} t}\right ]
\]