70.19.11 problem 11

Internal problem ID [19021]
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 : Thursday, October 02, 2025 at 03:37:02 PM
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*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 19

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-4*diff(diff(diff(y(t),t),t),t)+6*diff(diff(y(t),t),t)-4*diff(y(t),t)+y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 23

ode=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; 
ic={y[0]==0,Derivative[1][y][0] == 1,Derivative[2][y][0] == 0,Derivative[3][y][0] == 1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{3} e^t t \left (2 t^2-3 t+3\right ) \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 4*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)) - 4*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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