11.9 problem 1(i)

Internal problem ID [6655]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 93
Problem number : 1(i)
Date solved : Thursday, October 17, 2024 at 10:23:40 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y&=x^{2} {\mathrm e}^{-x} \end{align*}

11.9.1 Solved as higher order constant coeff ode

Time used: 0.120 (sec)

The characteristic equation is

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

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

\[ y_h(x)={\mathrm e}^{-x} c_1 +x \,{\mathrm e}^{-x} c_2 +x^{2} {\mathrm e}^{-x} c_3 \]

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

\begin{align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= x \,{\mathrm e}^{-x}\\ y_3 &= x^{2} {\mathrm e}^{-x} \end{align*}

This is higher order nonhomogeneous ODE. Let the solution be

\[ y = y_h + y_p \]

Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to

\[ y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 0 \]

Now the particular solution to the given ODE is found

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

The particular solution is now found using the method of undetermined coefficients.

Looking at the RHS of the ode, which is

\[ x^{2} {\mathrm e}^{-x} \]

Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is

\[ [\{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x}\}] \]

While the set of the basis functions for the homogeneous solution found earlier is

\[ \{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x}\} \]

Since \({\mathrm e}^{-x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes

\[ [\{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, x^{3} {\mathrm e}^{-x}\}] \]

Since \(x \,{\mathrm e}^{-x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes

\[ [\{x^{2} {\mathrm e}^{-x}, x^{3} {\mathrm e}^{-x}, x^{4} {\mathrm e}^{-x}\}] \]

Since \(x^{2} {\mathrm e}^{-x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes

\[ [\{x^{3} {\mathrm e}^{-x}, x^{4} {\mathrm e}^{-x}, x^{5} {\mathrm e}^{-x}\}] \]

Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set.

\[ y_p = A_{1} x^{3} {\mathrm e}^{-x}+A_{2} x^{4} {\mathrm e}^{-x}+A_{3} x^{5} {\mathrm e}^{-x} \]

The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives

\[ 24 A_{2} x \,{\mathrm e}^{-x}+60 A_{3} x^{2} {\mathrm e}^{-x}+6 A_{1} {\mathrm e}^{-x} = x^{2} {\mathrm e}^{-x} \]

Solving for the unknowns by comparing coefficients results in

\[ \left [A_{1} = 0, A_{2} = 0, A_{3} = {\frac {1}{60}}\right ] \]

Substituting the above back in the above trial solution \(y_p\), gives the particular solution

\[ y_p = \frac {x^{5} {\mathrm e}^{-x}}{60} \]

Therefore the general solution is

\begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-x} c_1 +x \,{\mathrm e}^{-x} c_2 +x^{2} {\mathrm e}^{-x} c_3\right ) + \left (\frac {x^{5} {\mathrm e}^{-x}}{60}\right ) \\ \end{align*}

11.9.2 Maple step by step solution

11.9.3 Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

11.9.4 Maple dsolve solution

Solving time : 0.007 (sec)
Leaf size : 24

dsolve(diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)+3*diff(y(x),x)+y(x) = exp(-x)*x^2, 
       y(x),singsol=all)
 

\[ y = {\mathrm e}^{-x} \left (\frac {1}{60} x^{5}+c_1 +c_2 x +x^{2} c_3 \right ) \]

11.9.5 Mathematica DSolve solution

Solving time : 0.011 (sec)
Leaf size : 34

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

\[ y(x)\to \frac {1}{60} e^{-x} \left (x^5+60 c_3 x^2+60 c_2 x+60 c_1\right ) \]