76.13.29 problem 29

Internal problem ID [17540]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 29
Date solved : Monday, March 31, 2025 at 04:16:52 PM
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 (0\right )&=2\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 14

ode:=9*diff(diff(y(x),x),x)-12*diff(y(x),x)+4*y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {2 x}{3}} \left (2-\frac {7 x}{3}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 21

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

\[ y(x)\to \frac {1}{3} e^{2 x/3} (6-7 x) \]

✓ Sympy. Time used: 0.161 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 12*Derivative(y(x), x) + 9*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (2 - \frac {7 x}{3}\right ) e^{\frac {2 x}{3}} \]