76.19.4 problem 4

Internal problem ID [17649]
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 : 4
Date solved : Thursday, March 13, 2025 at 10:45:56 AM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 9.519 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.160 (sec). Leaf size: 10

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

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