12.18.15 problem section 9.2, problem 15

Internal problem ID [2129]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant coefficient. Page 483
Problem number : section 9.2, problem 15
Date solved : Tuesday, March 04, 2025 at 01:50:37 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 )&=-2\\ y^{\prime \prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 23

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) = -2, (D@@2)(y)(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {5 \,{\mathrm e}^{2 x}}{4}-\frac {9 \sin \left (2 x \right )}{4}+\frac {3 \cos \left (2 x \right )}{4} \]

✓ Mathematica. Time used: 0.003 (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] ==-2,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.202 (sec). Leaf size: 27

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): -2, Subs(Derivative(y(x), (x, 2)), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {5 e^{2 x}}{4} - \frac {9 \sin {\left (2 x \right )}}{4} + \frac {3 \cos {\left (2 x \right )}}{4} \]