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6: Power Series and Laplace Transforms
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- 6.1: Review of Power Series
- Before we go on to solving differential equations using power series, it would behoove you to go back to you calculus notes and review power series. There is one topic that was a small detail in first year calculus, but will be a main issue for solving differential equations. This is the technique of changing the index.
- 6.2: Series Solutions to Second Order Linear Differential Equations
- We have fully investigated solving second order linear differential equations with constant coefficients. Now we will explore how to find solutions to second order linear differential equations whose coefficients are not necessarily constant.
- 6.3: Series Solutions and Convergence
- In the last section, we saw how to find series solutions to second order linear differential equations. We did not investigate the convergence of these series. In this discussion, we will derive an alternate method to find series solutions. We will also learn how to determine the radius of convergence of the solutions just by taking a quick glance of the differential equation.
- 6.4: Regular Singular Points
- When we worked out series solutions to differential equations in previous discussions, we always assumed that x0 was an ordinary point, that is p and q converged to their Taylor Series expansions. What happens if the point is a singular point?
- 6.5: Euler Equations
- In this section, we will investigate the solutions of the most simple type of differential equations with regular singular points .
- 6.6: The Laplace Transform
- We have seen many techniques of solving differential equations that involve using a substitution. There is a special type of substitution, called an integral transform that simplifies the task of solving differential equations.
- 6.7: Using the Laplace Transform to Solve Initial Value Problems
- Now that we know how to find a Laplace transform, it is time to use it to solve differential equations. The key feature of the Laplace transform that makes it a tool for solving differential equations is that the Laplace transform of the derivative of a function is an algebraic expression rather than a differential expression.
- 6.8: Step Functions
- In this discussion, we will investigate piecewise defined functions and their Laplace Transforms. We start with the fundamental piecewise defined function, the Heaviside function.
- 6.9: Discontinuous Forcing
- In the last section we looked at the Heaviside function its Laplace transform. Now we will use this tool to solve differential equations.