2.9.1 Problem 1

Internal problem ID [18583]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter VII. Linear equations of order higher than the first. section 63. Problems at page 196
Problem number : 1
Date solved : Saturday, February 22, 2025 at 09:30:38 PM
CAS classification : [[_3rd_order, _missing_x]]

Solve

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

Solved as higher order constant coeff ode

Time used: 0.025 (sec)

The characteristic equation is

\[ \lambda ^{3}-\lambda ^{2}-\lambda +1 = 0 \]

The roots of the above equation are

\begin{align*} \lambda _1 &= -1\\ \lambda _2 &= 1\\ \lambda _3 &= 1 \end{align*}

Therefore the homogeneous solution is

\[ y_h(x)={\mathrm e}^{-x} c_1 +{\mathrm e}^{x} c_2 +x \,{\mathrm e}^{x} c_3 \]

The fundamental set of solutions for the homogeneous solution are the following

\begin{align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= {\mathrm e}^{x}\\ y_3 &= x \,{\mathrm e}^{x} \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{3}-r^{2}-r +1=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial and corresponding multiplicities}\hspace {3pt} \\ {} & {} & r =\left [\left [-1, 1\right ], \left [1, 2\right ]\right ] \\ \bullet & {} & \textrm {Solution from}\hspace {3pt} r =-1 \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-x} \\ \bullet & {} & \textrm {1st solution from}\hspace {3pt} r =1 \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{x} \\ \bullet & {} & \textrm {2nd solution from}\hspace {3pt} r =1 \\ {} & {} & y_{3}\left (x \right )=x \,{\mathrm e}^{x} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=\mathit {C1} y_{1}\left (x \right )+\mathit {C2} y_{2}\left (x \right )+\mathit {C3} y_{3}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions and simplify}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{-x} \mathit {C1} +{\mathrm e}^{x} \left (\mathit {C3} x +\mathit {C2} \right ) \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 19

dsolve(diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-diff(y(x),x)+y(x) = 0,y(x),singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +{\mathrm e}^{x} \left (c_3 x +c_{2} \right ) \]

✓Mathematica DSolve solution

Solving time : 0.003 (sec)
Leaf size : 25

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

\[ y(x)\to c_1 e^{-x}+e^x (c_3 x+c_2) \]

✓Sympy solution

Solving time : 0.191 (sec)
Leaf size : 15

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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

Eq(y(x), C3*exp(-x) + (C1 + C2*x)*exp(x))
 

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