89.11.26 problem 26

Internal problem ID [24551]
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 117
Problem number : 26
Date solved : Thursday, October 02, 2025 at 10:46:03 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} -4 y^{\prime }+y^{\prime \prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 8

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

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 6

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

\begin{align*} y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 20

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

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