55.27.1 problem 1

Internal problem ID [13774]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power Functions. page 213
Problem number : 1
Date solved : Thursday, October 02, 2025 at 08:06:35 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

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

\[ y = c_1 \sin \left (\sqrt {a}\, x \right )+c_2 \cos \left (\sqrt {a}\, x \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to c_1 \cos \left (\sqrt {a} x\right )+c_2 \sin \left (\sqrt {a} x\right ) \end{align*}

✓ Sympy. Time used: 0.046 (sec). Leaf size: 26

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)
 

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