76.19.11 problem 11

Internal problem ID [17735]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 11:01:59 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.423 (sec). Leaf size: 19

dsolve([diff(y(t),t$4)-4*diff(y(t),t$3)+6*diff(y(t),t$2)-4*diff(y(t),t)+y(t)=0,y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{t} t \left (2 t^{2}-3 t +3\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 23

DSolve[{D[y[t],{t,4}]-4*D[y[t],{t,3}]+6*D[y[t],{t,2}]-4*D[y[t],t]+y[t]==0,{y[0]==0,Derivative[1][y][0] == 1,Derivative[2][y][0] == 0,Derivative[3][y][0] == 1}},y[t],t,IncludeSingularSolutions -> True]
 

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