43.8.1 problem 1(c)

Internal problem ID [8934]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 79
Problem number : 1(c)
Date solved : Tuesday, September 30, 2025 at 06:00:18 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-4 y^{\prime }&=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: 0.054 (sec). Leaf size: 10

ode:=diff(diff(diff(y(x),x),x),x)-4*diff(y(x),x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sinh \left (2 x \right )}{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 69

ode=D[y[x],{x,3}]-4*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 {e^{-\frac {x}{\sqrt [3]{2}}} \left (e^{\frac {3 x}{\sqrt [3]{2}}}+\sqrt {3} \sin \left (\frac {\sqrt {3} x}{\sqrt [3]{2}}\right )-\cos \left (\frac {\sqrt {3} x}{\sqrt [3]{2}}\right )\right )}{3\ 2^{2/3}} \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*Derivative(y(x), 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)
 

\[ y{\left (x \right )} = \frac {e^{2 x}}{4} - \frac {e^{- 2 x}}{4} \]