43.8.2 problem 2(c)

Internal problem ID [8935]
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 : 2(c)
Date solved : Tuesday, September 30, 2025 at 06:00:18 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (5\right )}-y^{\prime \prime \prime \prime }-y^{\prime }+y&=0 \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 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.086 (sec). Leaf size: 28

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-diff(diff(diff(diff(y(x),x),x),x),x)-diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0, (D@@4)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x}}{8}+\frac {\left (-2 x +5\right ) {\mathrm e}^{x}}{8}+\frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 34

ode=D[y[x],{x,5}]-D[y[x],{x,4}]-D[y[x],x]+y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0,Derivative[4][y][0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (-2 e^x x+e^{-x}+5 e^x-2 \sin (x)+2 \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - Derivative(y(x), x) - Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0, Subs(Derivative(y(x), (x, 3)), x, 0): 0, Subs(Derivative(y(x), (x, 4)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {5}{8} - \frac {x}{4}\right ) e^{x} - \frac {\sin {\left (x \right )}}{4} + \frac {\cos {\left (x \right )}}{4} + \frac {e^{- x}}{8} \]