68.1.59 problem 81

Internal problem ID [17126]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 81
Date solved : Thursday, October 02, 2025 at 01:44:07 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+\frac {25 y^{\prime \prime }}{2}-5 y^{\prime }+\frac {629 y}{16}&=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 )&=-1 \\ y^{\prime \prime \prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 41

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+25/2*diff(diff(y(x),x),x)-5*diff(y(x),x)+629/16*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = -1, (D@@3)(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (74 \cos \left (3 x \right )+20 \sin \left (3 x \right )\right ) {\mathrm e}^{-\frac {x}{2}}}{208}-\frac {37 \left (\cos \left (2 x \right )-\frac {3 \sin \left (2 x \right )}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{104} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 47

ode=D[y[x],{x,4}]+25/2*D[y[x],{x,2}]-5*D[y[x],x]+629/16*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1,Derivative[2][y][0] ==-1,Derivative[3][y][0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{208} e^{-x/2} \left (111 e^x \sin (2 x)+20 \sin (3 x)-74 e^x \cos (2 x)+74 \cos (3 x)\right ) \end{align*}

✓ Sympy. Time used: 0.265 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(629*y(x)/16 - 5*Derivative(y(x), x) + 25*Derivative(y(x), (x, 2))/2 + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1, Subs(Derivative(y(x), (x, 2)), x, 0): -1, Subs(Derivative(y(x), (x, 3)), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {111 \sin {\left (2 x \right )}}{208} - \frac {37 \cos {\left (2 x \right )}}{104}\right ) e^{\frac {x}{2}} + \left (\frac {5 \sin {\left (3 x \right )}}{52} + \frac {37 \cos {\left (3 x \right )}}{104}\right ) e^{- \frac {x}{2}} \]