8.9.4 problem 4

Internal problem ID [2570]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:47:18 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 11

ode:=4*diff(diff(y(t),t),t)-4*diff(y(t),t)+y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 3 \,{\mathrm e}^{\frac {t}{2}} t \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 15

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

\begin{align*} y(t)&\to 3 e^{t/2} t \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 10

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

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