49.9.6 problem 2

Internal problem ID [7655]
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 : Monday, January 27, 2025 at 03:09:05 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*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 39

dsolve([diff(y(x),x$3)+y(x)=0,y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0],y(x), singsol=all)
 

\[ y = \frac {\left ({\mathrm e}^{\frac {3 x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}+{\mathrm e}^{\frac {3 x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-1\right ) {\mathrm e}^{-x}}{3} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 59

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

\[ y(x)\to \frac {1}{3} e^{-x} \left (\sqrt {3} e^{3 x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+e^{3 x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )-1\right ) \]