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 : Wednesday, February 05, 2025 at 03:45:45 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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t$4)+2*diff(x(t),t$2)+x(t)=exp(2*t),x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 0, (D@@3)(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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