Calc 3 Lecture Notes Section 15.4 Page 9 of 10
Section 15.4: Power Series Solutions of Differential Equations
Big idea: You can solve differential equations with power series.
Big skill: You should be able to solve differential equations with power series.
When the coefficients of a second-order differential equation are not constants, but instead are polynomial functions of the independent variable, like , then one way to solve the equation is using a power series technique:
· Assume a power series form of the solution
· Plug the power series into the equation and equate like terms
· Obtain a recurrence relation for the coefficients of higher-order terms in terms of the first two coefficients, then try to obtain an explicit function for the coefficients based on the term index.
· Write the final answer in terms of those coefficients.
Practice:
1. Find the general solution of using techniques from 15.1, then solve it using the power series technique.
2. Find the general solution of using the power series technique.
3. Find the general solution of Airy’s equation using the power series technique. Airy’s equation arises in the description of the diffraction of light from a circular aperture.
4. The Bessel functions of the first kind of order p, Jp, are the solutions of the differential equation , where p is a nonnegative integer. Show that is the solution to the above differential equation. Bessel functions can be used to describe the vibrations of a circular drum head.
5. Hermite’s equation is for integer values of k ³ 0. Show that one of the series solutions is a polynomial of degree k. When the leading coefficient is chosen to be 2k, then the polynomial solutions for different k are called the Hermite polynomials Hk(x). Find the first few Hermite polnomials.
6. Chebyshev polynomials are polynomial solutions of the equation for some integer k ³ 0. The polynomials Tk(x) obtained for different values of k are called Chebyshev polynomials. Find the first few Chebyshev polynomials. Notice that these polynomials can be obtained trigonometrically from the equation .
Some Multiple Angle Formulas for Trigonometric Functions
By Kevin Mirus, Madison Area Technical College