89.13.1 problem 1

Internal problem ID [24584]
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 127
Problem number : 1
Date solved : Thursday, October 02, 2025 at 10:46:15 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+a^{2} y-2 a y^{\prime }+b^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+a^2*y(x)-2*a*diff(y(x),x)+b^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{a x} \left (c_1 \sin \left (b x \right )+c_2 \cos \left (b x \right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to e^{x (a-i b)} \left (c_2 e^{2 i b x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} e^{x \left (a - i b\right )} + C_{2} e^{x \left (a + i b\right )} \]