64.10.40 problem 40

Internal problem ID [13367]
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 : 40
Date solved : Wednesday, March 05, 2025 at 09:49:16 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }-8 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 21

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

\[ y(x)\to e^{2 x}-\sin (2 x)+\cos (2 x) \]

✓ Sympy. Time used: 0.190 (sec). Leaf size: 17

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

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