23.4.2 problem 2

Internal problem ID [6304]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 02:45:35 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a y \end{align*}

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

ode:=diff(diff(y(x),x),x) = a*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{\sqrt {a}\, x}+c_2 \,{\mathrm e}^{-\sqrt {a}\, x} \]

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

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

\begin{align*} y(x)&\to c_1 e^{\sqrt {a} x}+c_2 e^{-\sqrt {a} x} \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 22

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

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