89.12.23 problem 23

Internal problem ID [24577]
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 : 23
Date solved : Thursday, October 02, 2025 at 10:46:12 PM
CAS classification : [[_3rd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 20

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

\[ y = \left (2 \,{\mathrm e}^{3 x}+3 x -2\right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to e^{-x} \left (3 x+2 e^{3 x}-2\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 17

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

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