5.2.2 problem 9

Internal problem ID [1473]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 4.2, Higher order linear differential equations. Constant coefficients. page 180
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:34:28 AM
CAS classification : [[_3rd_order, _missing_x]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 49

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+3*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\left (2^{{1}/{3}}-1\right ) x} \left (\left (\sin \left (\frac {\sqrt {3}\, 2^{{1}/{3}} x}{2}\right ) c_2 +\cos \left (\frac {\sqrt {3}\, 2^{{1}/{3}} x}{2}\right ) c_3 \right ) {\mathrm e}^{\frac {3 x 2^{{1}/{3}}}{2}}+c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 87

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

\begin{align*} y(x)&\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,3\right ]\right ) \end{align*}

✓ Sympy. Time used: 0.196 (sec). Leaf size: 71

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

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