8. Variable Coefficient Ordinary Differential Equations

The entire course of study has involved the evolution of solution strategies; beginning with the solution of simple (e.g. first order) ODEs, then constant coefficient homogeneous and non-homogeneous ODEs, and now culminating with the solution of variable coefficient differential equations.

Such ODEs are realized in two ways: as a sub-step in the solution of Partial Differential Equations (PDEs), or in the solution of increasingly complex physical problems.  Consider, for example, the spring-mass-damper system described by the ODE:
        m*y'' + c*y' + k*y = f(t)

where m, c and k are coefficients corresponding to the mass, damping and spring components of the system.  It is easy to envision a scenario where these coefficients are not constant - that is, they change as a function of time.  The mass may be a container full of liquid, which may drain or evaporate in time.  Further, the spring and damper components of the system may heat up in time which may also affect their performance.  The resulting ODE then becomes:
        m(t)*y'' + c(t)*y' + k(t)*y = f(t)

In this section of the course, we shall consider a number of strategies for solving variable coefficient ODEs like the one presented above.  The strategies considered include:

  • Reduction of Order (to first order)
  • Change of Independent Variable (to create a constant coefficient ODE)
  • Solution by Series (a brute force method)

    •  

    8.1.    Reduction to First Order

    Little will be said for this solution method; the first order linear (FOL) formula can be used to solve the problem if the original ODE can be reduced to a first order equation.  Recall, that the FOL formula had no constraint for for constant coefficients.
     

    8.2.    Change of Independent Variable

    The general idea of this strategy is summarized as:
  • In the var. coef. ODE, replace the original independent variable, x, with a new one, z=f(x)

    •   ** The objective is to form a constant coefficient ODE (in y(z)) with this substitution
  • Solve this constant coefficient ODE for y(z)
  • Substitute z=f(z) to obtain the required solution, y(x)


    • One of the most common examples where this strategy applies is in the solution of the Cauchy-Euler (or simply Euler) ODE:
        c_2*x^2*y'' + c_1*x^2*y' + c_0*y = f(x)

    This ODE is characterized by an x^n in the coefficient of the n'th order derivative term.  It is not limited to a second order ODE - i.e. a 10th order ODE of similar form is still an Euler ODE.

    As suggested by the example given here, this ODE can always be reduced to a constant coefficient ODE using the substitution:     z=ln(x)
     

    8.3.    Solutions Using Power Series

    Power series (PS) are often useful tools in the solution of ODEs - especially those with variable coefficients.  This solution strategy is usually more involved than many other methods, but often represents the only way in which an analytical solution may be obtained.  Some things to consider before using PS solution strategies:
  • In review of general power series, the question of convergence comes to light - this issue is still significant when power series are used in the solution of ODEs.  Indeed, depending on the form of the ODE, the PS may only converge over a finite range of the solution domain and the determined solution will only be valid in that range.
  • It is not easy to 'see' the form of the PS solution.  For instance, solutions containing sines and cosines may be represented as PS in terms of even and odd powers of the independent variables, but they are not easy to recognize in that form.


    • There are two main PS that arise in the solution of variable coefficient ODEs - the choice of which to use (as will be seen below) depends upon the coefficients and the form of the ODE itself:

  • Taylor (or Maclaurin) Series:  terms of the form a_n*x^n
  • Frobenius Series: terms of the form a_n*x^(c+n)


    • In each case, there are a fixed set of steps that are applied in the solution process - their origin will become apparent as we begin to examine the examples given below:

  • Check the form of the ODE to determine which PS applies (Maclaurin/Frobenius)
  • Apply the series to the ODE
  • Expand and subsequently combine terms to a single summation

    •     This requires unification of summation indices, and powers of the independent variable
  • Solve for the coefficients of the series (i.e. the a_n values)

    • 8.3.1.    Algebra of Series (Review)

    A brief review of the algebra of PS is warranted before moving to apply them in the solution of ODEs.  This review is available here.

    8.3.2.    Existence and Convergence of Series Solutions

    As highlighted above, the use of PS introduces the possibility that the ODE solution may not exist (i.e. the series never converges) or that its convergence is limited to a finite range.  These ideas are investigated here, and set the stage for the classification of ODEs for which the PS solution strategy may be applied.  Moreover, they are also useful to determine which of the Maclaurin or Frobenius series will yield a convergent solution, and their respective intervals of convergence.

    A Maclaurin Series Example

    A Frobenius Series Example:  Bessel's Equation