89.12.27 problem 27

Internal problem ID [24581]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 121
Problem number : 27
Date solved : Thursday, October 02, 2025 at 10:46:14 PM
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 )&=-2 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 14

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

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

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

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

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

✓ Sympy. Time used: 0.092 (sec). Leaf size: 10

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 = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

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