6.18.22 problem section 9.2, problem 22

Internal problem ID [2136]
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 22
Date solved : Tuesday, September 30, 2025 at 05:24:19 AM
CAS classification : [[_3rd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=-3 \\ y^{\prime \prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 22

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

\[ y = 3 \,{\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{\frac {x}{2}}-2 \,{\mathrm e}^{\frac {x}{2}} x \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to -\frac {2}{21} e^{-x/4} \left (9 e^{5 x/4}+13 \sqrt {3} \sin \left (\frac {\sqrt {3} x}{4}\right )-51 \cos \left (\frac {\sqrt {3} x}{4}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (1 - 2 x\right ) e^{\frac {x}{2}} + 3 e^{- \frac {x}{2}} \]