14.16.1 problem 15

Internal problem ID [2671]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 15
Date solved : Monday, January 27, 2025 at 06:06:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+4 y&={\mathrm e}^{2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.837 (sec). Leaf size: 21

dsolve([diff(y(t),t$2)-5*diff(y(t),t)+4*y(t)=exp(2*t),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = 2 \,{\mathrm e}^{t}-\frac {{\mathrm e}^{4 t}}{2}-\frac {{\mathrm e}^{2 t}}{2} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 22

DSolve[{D[y[t],{t,2}]-5*D[y[t],t]+4*y[t]==Exp[2*t],{y[0]==1,Derivative[1][y][0] ==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {1}{2} e^t \left (e^t+e^{3 t}-4\right ) \]