Preface

---

This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

We recall the definition of a root multiplicity. A real or complex number α is called a root of multiplicity k of the polynomial p(x) if there is a polynomial s(x) such that \( s(\alpha ) \ne 0 \) and \( p(x) = \left( x- \alpha \right)^k s(x) . \) If k=1, then α is called a simple root. If \( k \ge 2, \) then α is called a multiple root.

Reduction of order for constant coefficient equations

---

We start with constant coefficient linear homogeneous differential equations. Suppose that in a constant coefficient linear differential operator

the operator \( L \left[ \texttt{D} \right] = a_n \texttt{D}^n + a_{n-1} \texttt{D}^{n-1} + \cdots + a_1 \texttt{D} + a_0 ,\) with \( \texttt{D} = {\text d}/{\text d}x \quad\mbox{and} \quad a_n \ne 0, \) has repeated factors. That is, the characteristic polynomial \( L \left( \lambda \right) = a_n \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_0 \) has repeated roots. For example, the second order constant coefficient linear differential operator \( L \left[ \texttt{D} \right] = a\,\texttt{D}^2 + b\,\texttt{D} +c \) has a repeated factor if and only if the corresponding characteristic equation \( a\,\lambda^2 + b\,\lambda + c =0 \) has a double root: In other words, the quadratic polynomial can be factored \( a\,\lambda^2 + b\,\lambda + c = a \left( \lambda - \lambda_1 \right)^2 \) if and only if its discriminant \( b^2 - 4ac \) is zero. In this case we have only one solution of exponential form: for the differential equation

To find another linearly independent solution to the above homogeneous equation, we use the method of reduction of order, credited to Jacob Bernoulli (1655--1705). Setting

we have Upon substituting the Bernoulli form into the differential equation, we get The latter can be divided by nonzero term \( a\,y_1 (x) = a\,e^{-bx/(2a)} . \) This yields \( v'' (x) =0, \) and after integration, we obtain \( v (x) = C_1 + C_2 x, \) with arbitrary constants C₁ and C₂. Finally, substituting for v(x), we define the general solution Functions \( y_1 (x) = e^{-bx/(2a)} \quad \mbox{and} \quad y_2 (x) = x\,e^{-bx/(2a)} \) are linearly independent because their Wronskian Therefore, these two functions form a fundamental set of solutions whenever γ is. ■

This method can be extended for arbitrary case when a linear constant coefficient differential operator has a multiple \( \left( \texttt{D} - \gamma \right)^m \) of multiplicity m. Then to such multiple correspond m linearly independent solutions:

The strict derivation of the above statement is made in annihilation section.

Theorem: Let \( a_0 , a_1 , \ldots , a_n \) be n+1 real (or complex) numbers with \( a_n \ne 0 , \) and y(x) be a n times continuously differentiable function on some interval |a,b|. Then y(x) is a solution of the n-th order linear differential equation with constant coefficients

if and only if where \( \lambda_1 , \lambda_2 , \ldots , \lambda_m \) are distinct roots of the characteristic polynomial \( \sum_{k=1}^n a_k \lambda^k \) with multiplicities \( m_1 , m_2 , \ldots , m_m , \) respectively, and \( P_k (x) \) is a polynomial of degree \( m_k -1 . \) ■

  We consider a partcular case of second order differential equation---the motion for undriven damped harmonic oscillator (coefficients μ and ω₀ > 0 are assumed constants):

subject to the initial conditions We show that its solution can be obtained from the solution by taking the limit when \( \lambda_1 \ \to \, \lambda_2 . \) Here are two distinct roots of the characteristic equation \( \lambda^2 + 2\mu\,\lambda + \omega_0^2 =0 . \) Let assume, for simplicity, that \( \eta = \left( \mu^2 - \omega_0^2 \right)^{1/2} \ge 0 , \) that is \( \mu \ge \omega_0 . \) Then rewriting the solution in the equivalent form: and taking the limit as \( \eta \, \to \, 0, \) one gets

 

Example: Homogeneous constant coefficient equation (one double root) Example: Solution of the initial value problem (one double root) Example: Solution of the fourth order linear differential equation (one triple root)

 

Reduction of order for variable coefficient equations

---

The reduction of order technique, which applies to arbitrary linear differential equations, allows us to go beyond equations with constant coefficients, provided that we already know one solution. For sake of clarity, we start with a second order linear differential equation with variable coefficients:

where p(x) and q(x) are some continuous functions on some interval |a,b|. Suppose that we know one its solution \( y = y_1 (x) \ne 0. \) This means that Then we seek another linearly independent solution in the form \( y(x) = v(x)\, y_1 (x) , \) where unknown function v(x) s determined upon substitution of this form into the given differential equation. First, we use the product rule to obtain This implies that we must have Since \( L \left[ \texttt{D} \right] y_1 \equiv 0 , \) the function v(x) must satisfy the equation This is a second order differential equation where dependent function is missing. If we set \( u= v' , \) we reduce it to a first order differential equation: Integrating both sides, we get This allows us to determine u(x) explicitly: where we dropped an arbitrary constant because we are after another linearly independent solution. Next integration yields Example: Solution of the second order order variable coefficient equation

 

Return to Mathematica page

Return to the main page (APMA0330)
Return to the Part 1 (Plotting)
Return to the Part 2 (First Order ODEs)
Return to the Part 3 (Numerical Methods)
Return to the Part 4 (Second and Higher Order ODEs)
Return to the Part 5 (Series and Recurrences)
Return to the Part 6 (Laplace Transform)