2.8.1 Problem 1

Internal problem ID [19762]
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 : Friday, November 28, 2025 at 06:51:28 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: 0.051 (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

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

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. Time used: 0.002 (sec). Leaf size: 19

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

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

Maple trace

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

Maple step by step

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to c_1 e^{-x}+e^x (c_3 x+c_2) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 15

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)
 

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