3.2.22 problem problem 48

Internal problem ID [956]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.3, Higher-Order Linear Differential Equations. Homogeneous Equations with Constant Coefficients. Page 300
Problem number : problem 48
Date solved : Tuesday, September 30, 2025 at 04:19:58 AM
CAS classification : [[_3rd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.102 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.106 (sec). Leaf size: 26

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

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