14.1.15 problem Problem 21

Internal problem ID [3572]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.2, Basic Ideas and Terminology. page 21
Problem number : Problem 21
Date solved : Tuesday, September 30, 2025 at 06:45:57 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-2*a*diff(y(x),x)+(a^2+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: 0.011 (sec). Leaf size: 31

ode=D[y[x],{x,2}]-2*a*D[y[x],x]+(a^2+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: 0.134 (sec). Leaf size: 22

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

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