13.9.6 problem 7

Internal problem ID [2392]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:00:09 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }-12 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\pi \right )&=0\\ y^{\prime }\left (\pi \right )&=2 \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 19

ode:=9*diff(diff(y(t),t),t)-12*diff(y(t),t)+4*y(t) = 0; 
ic:=y(Pi) = 0, D(y)(Pi) = 2; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 24

ode=9*D[y[t],{t,2}]-12*D[y[t],t]+4*y[t]==0; 
ic={y[Pi]==0,Derivative[1][y][Pi]==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-\frac {2}{3} (\pi -t)} (2 t-2 \pi ) \]

✓ Sympy. Time used: 0.219 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 12*Derivative(y(t), t) + 9*Derivative(y(t), (t, 2)),0) 
ics = {y(pi): 0, Subs(Derivative(y(t), t), t, pi): 2} 
dsolve(ode,func=y(t),ics=ics)
 

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