9.1.2 problem problem 39

Internal problem ID [929]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.2, Higher-Order Linear Differential Equations. General solutions of Linear Equations. Page 288
Problem number : problem 39
Date solved : Saturday, March 29, 2025 at 10:34:53 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{\frac {x}{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=4*diff(diff(y(x),x),x)-4*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {x}{2}} \left (c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 20

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

\[ y(x)\to e^{x/2} (c_2 x+c_1) \]

✓ Sympy. Time used: 0.143 (sec). Leaf size: 12

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

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