89.12.25 problem 25

Internal problem ID [24579]
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 121
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:46:13 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+2 y^{\prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=3 \\ y^{\prime \prime }\left (0\right )&=-5 \\ y^{\prime \prime \prime }\left (0\right )&=9 \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 18

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+3*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 3, (D@@2)(y)(0) = -5, (D@@3)(y)(0) = 9]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2-{\mathrm e}^{-x}-{\mathrm e}^{-2 x} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 21

ode=D[y[x],{x,4}]+3*D[y[x],{x,3}]+2*D[y[x],{x,2}] ==0; 
ic={y[0]==0,Derivative[1][y][0] ==3,Derivative[2][y][0] ==-5,Derivative[3][y][0] ==9}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -e^{-2 x}-e^{-x}+2 \end{align*}

✓ Sympy. Time used: 0.075 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 3, Subs(Derivative(y(x), (x, 2)), x, 0): -5, Subs(Derivative(y(x), (x, 3)), x, 0): 9} 
dsolve(ode,func=y(x),ics=ics)
 

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