64.10.41 problem 41

Internal problem ID [13376]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.2. The homogeneous linear equation with constant coefficients. Exercises page 135
Problem number : 41
Date solved : Monday, March 31, 2025 at 07:52:42 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-8\\ y^{\prime \prime }\left (0\right )&=-4 \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 22

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+4*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = -8, (D@@2)(y)(0) = -4; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (6 x -23\right ) {\mathrm e}^{2 x}}{9}+\frac {32 \,{\mathrm e}^{-x}}{9} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 27

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]+4*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-8,Derivative[2][y][0] ==-4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{9} e^{-x} \left (e^{3 x} (6 x-23)+32\right ) \]

✓ Sympy. Time used: 0.131 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -8, Subs(Derivative(y(x), (x, 2)), x, 0): -4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {2 x}{3} - \frac {23}{9}\right ) e^{2 x} + \frac {32 e^{- x}}{9} \]