84.40.5 problem 27.11
Internal
problem
ID
[22380]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
27.
Solutions
of
systems
of
linear
differential
equations
with
constant
coefficients
by
Laplace
transform.
Supplementary
problems
Problem
number
:
27.11
Date
solved
:
Sunday, October 12, 2025 at 05:52:22 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}u \left (t \right )-2 v \left (t \right )&=2\\ u \left (t \right )+\frac {d}{d t}v \left (t \right )&=5 \,{\mathrm e}^{2 t}+1 \end{align*}
With initial conditions
\begin{align*}
u \left (0\right )&=2 \\
D\left (u \right )\left (0\right )&=2 \\
v \left (0\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.421 (sec). Leaf size: 101
ode:=[diff(diff(u(t),t),t)-2*v(t) = 2, u(t)+diff(v(t),t) = 5*exp(2*t)+1];
ic:=[u(0) = 2, D(u)(0) = 2, v(0) = 1];
dsolve([ode,op(ic)]);
\begin{align*}
u \left (t \right ) &= \frac {10}{\left (2^{{1}/{3}}+2\right ) \left (4-2 \,2^{{1}/{3}}+2^{{2}/{3}}\right )}+\frac {10 \,{\mathrm e}^{2 t}}{\left (2^{{1}/{3}}+2\right ) \left (4-2 \,2^{{1}/{3}}+2^{{2}/{3}}\right )} \\
v \left (t \right ) &= -\frac {10}{\left (2^{{1}/{3}}+2\right ) \left (4-2 \,2^{{1}/{3}}+2^{{2}/{3}}\right )}+\frac {20 \,{\mathrm e}^{2 t}}{\left (2^{{1}/{3}}+2\right ) \left (4-2 \,2^{{1}/{3}}+2^{{2}/{3}}\right )} \\
\end{align*}
✓ Mathematica. Time used: 0.04 (sec). Leaf size: 1177
ode={D[u[t],{t,2}]-2*v[t]==2,u[t]+D[v[t],t]==5*Exp[2*t]+1};
ic={u[0]==2,Derivative[1][u][0] ==2,v[0]==1};
DSolve[{ode,ic},{u[t],v[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 11.015 (sec). Leaf size: 488
from sympy import *
t = symbols("t")
u = Function("u")
v = Function("v")
ode=[Eq(-2*v(t) + Derivative(u(t), (t, 2)) - 2,0),Eq(u(t) - 5*exp(2*t) + Derivative(v(t), t) - 1,0)]
ics = {u(0): 2, Subs(Derivative(u(t), t), t, 0): 2, v(0): 1}
dsolve(ode,func=[u(t),v(t)],ics=ics)
\[
\left [ u{\left (t \right )} = - \frac {10 \left (- 26781 \cdot 2^{\frac {2}{3}} + 4671 + 30107 \sqrt [3]{2}\right ) e^{2 t} \sin ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 80879 \sqrt [3]{2} + 12132 \cdot 2^{\frac {2}{3}} + 81788\right )} - \frac {10 \left (- 26781 \cdot 2^{\frac {2}{3}} + 4671 + 30107 \sqrt [3]{2}\right ) e^{2 t} \cos ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 80879 \sqrt [3]{2} + 12132 \cdot 2^{\frac {2}{3}} + 81788\right )} + \frac {5 \sqrt [3]{2} e^{2 t}}{3 \left (\sqrt [3]{2} + 2\right )} - \frac {5 \left (-37568 + 11323 \cdot 2^{\frac {2}{3}} + 15994 \sqrt [3]{2}\right ) \sin ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 80879 \sqrt [3]{2} + 12132 \cdot 2^{\frac {2}{3}} + 81788\right )} - \frac {5 \left (-37568 + 11323 \cdot 2^{\frac {2}{3}} + 15994 \sqrt [3]{2}\right ) \cos ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 80879 \sqrt [3]{2} + 12132 \cdot 2^{\frac {2}{3}} + 81788\right )} + \frac {1 - \sqrt [3]{2}}{3}, \ v{\left (t \right )} = \frac {5 \left (- 1434 \cdot 2^{\frac {2}{3}} + 554 + 1273 \sqrt [3]{2}\right ) e^{2 t} \sin ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 1753 \cdot 2^{\frac {2}{3}} + 716 \sqrt [3]{2} + 1748\right )} + \frac {5 \left (- 31 \sqrt [3]{2} + 22 + 18 \cdot 2^{\frac {2}{3}}\right ) e^{2 t} \cos ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 32 \sqrt [3]{2} + 4 + 31 \cdot 2^{\frac {2}{3}}\right )} + \frac {5 e^{2 t}}{3 \left (\sqrt [3]{2} + 2\right )} + \frac {\left (-4212 + 321 \sqrt [3]{2} + 2632 \cdot 2^{\frac {2}{3}}\right ) \sin ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 1753 \cdot 2^{\frac {2}{3}} + 716 \sqrt [3]{2} + 1748\right )} + \frac {\left (- 64 \cdot 2^{\frac {2}{3}} + 24 + 33 \sqrt [3]{2}\right ) \cos ^{2}{\left (\frac {\sqrt [3]{2} \sqrt {3} t}{2} \right )}}{3 \left (- 32 \sqrt [3]{2} + 4 + 31 \cdot 2^{\frac {2}{3}}\right )} - \frac {- \sqrt [3]{2} \left (\sqrt [3]{2} + 2\right ) + 1 + 3 \sqrt [3]{2}}{3 \left (\sqrt [3]{2} + 2\right )}\right ]
\]