44.25.5 problem 4(a)

Internal problem ID [9463]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section A, Drill exercises. Page 309
Problem number : 4(a)
Date solved : Tuesday, September 30, 2025 at 06:19:05 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method

✓ Maple. Time used: 0.103 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{\frac {5 t}{2}} \left (3 y \left (0\right ) \cosh \left (\frac {3 t}{2}\right )+\sinh \left (\frac {3 t}{2}\right ) \left (2 y^{\prime }\left (0\right )-5 y \left (0\right )\right )\right )}{3} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

ode=D[y[t],{t,2}]-5*D[y[t],t]+4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^t \left (c_2 e^{3 t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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