43.9.6 problem 2

Internal problem ID [8941]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 83
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:00:22 PM
CAS classification : [[_3rd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ y^{\prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 40

ode:=diff(diff(diff(y(x),x),x),x)+y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 59

ode=D[y[x],{x,3}]+y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{-x} \left (\sqrt {3} e^{3 x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+e^{3 x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )-1\right ) \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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