7.19.10 problem 36

Internal problem ID [550]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 36
Date solved : Tuesday, March 04, 2025 at 11:26:32 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime \prime \prime }+2 x^{\prime \prime }+x&={\mathrm e}^{2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=0\\ x^{\prime }\left (0\right )&=0\\ x^{\prime \prime }\left (0\right )&=0\\ x^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.211 (sec). Leaf size: 29

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+2*diff(diff(x(t),t),t)+x(t) = exp(2*t); 
ic:=x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 0, (D@@3)(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = \frac {{\mathrm e}^{2 t}}{25}+\frac {\cos \left (t \right ) \left (-1+5 t \right )}{25}+\frac {\left (-14-5 t \right ) \sin \left (t \right )}{50} \]

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 35

ode=D[x[t],{t,4}]+2*D[x[t],{t,2}]+x[t]==Exp[2*t]; 
ic={x[0]==0,Derivative[1][x][0] ==0,Derivative[2][x][0] ==0,Derivative[3][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{50} \left (2 e^{2 t}-(5 t+14) \sin (t)+2 (5 t-1) \cos (t)\right ) \]

✓ Sympy. Time used: 0.190 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) - exp(2*t) + 2*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 4)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0, Subs(Derivative(x(t), (t, 2)), t, 0): 0, Subs(Derivative(x(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {t}{10} - \frac {7}{25}\right ) \sin {\left (t \right )} + \left (\frac {t}{5} - \frac {1}{25}\right ) \cos {\left (t \right )} + \frac {e^{2 t}}{25} \]