89.14.14 problem 14

Internal problem ID [24618]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 128
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:46:31 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+9 y^{\prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime }\left (\infty \right )&=1 \\ y^{\prime \prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 14

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+6*diff(diff(diff(y(x),x),x),x)+9*diff(diff(y(x),x),x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 0, D(y)(infinity) = 1, (D@@2)(y)(0) = 6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x \left (1-{\mathrm e}^{-3 x}\right ) \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 15

ode=D[y[x],{x,4}]+6*D[y[x],{x,3}]+9*D[y[x],{x,2}]==0; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[1][y][Infinity] ==1,Derivative[2][y][0] ==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x-e^{-3 x} x \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*Derivative(y(x), (x, 2)) + 6*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), x), x, oo): 1, Subs(Derivative(y(x), (x, 2)), x, 0): 6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - C_{4} + C_{4} e^{- 3 x} + x \left (\frac {3 C_{4}}{2} + \left (\frac {3 C_{4}}{2} - 1\right ) e^{- 3 x} + 1\right ) \]