Calc 3 Lecture Notes
Section 15.4
Page 1 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 y  x   x2 y  x   xy  x   0 , then one
way to solve the equation is using a power series technique:


Assume a power series form of the solution y  x    an x n
n 0



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 y  x   y  x   0 using techniques from 15.1, then solve it
using the power series technique.
Calc 3 Lecture Notes
Section 15.4
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Calc 3 Lecture Notes
Section 15.4
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2. Find the general solution of y  x   2 xy  x   2 y  x   0 using the power series
technique.
Calc 3 Lecture Notes
Section 15.4
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Calc 3 Lecture Notes
Section 15.4
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3. Find the general solution of Airy’s equation y  x   xy  x   0 using the power series
technique. Airy’s equation arises in the description of the diffraction of light from a
circular aperture.
Calc 3 Lecture Notes
Section 15.4
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Calc 3 Lecture Notes
Section 15.4
Page 7 of 10
4. The Bessel functions of the first kind of order p, Jp, are the solutions of the differential
equation x 2 y  xy   x 2  p 2  y  0 , where p is a nonnegative integer. Show that
 1
x2k  p
is the solution to the above differential equation. Bessel
2k  p
k ! k  p  !
k 0 2
functions can be used to describe the vibrations of a circular drum head.

J p  x  
k
Calc 3 Lecture Notes
Section 15.4
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5. Hermite’s equation is y  x   2 xy  x   2ky  x   0 for integer values of k  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.
Calc 3 Lecture Notes
Section 15.4
Page 9 of 10
6. Chebyshev polynomials are polynomial solutions of the equation
1  x2  y  x   xy  x   k 2 y  x   0 for some integer k  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 Tk  cos     cos  k  .
Calc 3 Lecture Notes
Section 15.4
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Some Multiple Angle Formulas for Trigonometric Functions
By Kevin Mirus, Madison Area Technical College
cos  2t   cos2  t   sin 2  t   2cos2  t  1
cos  3t   cos  2t  cos  t   sin  2t  sin  t 
  2 cos 2  t   1 cos  t   2sin 2  t  cos  t 
 2 cos3  t   cos  t   2 1  cos 2  t   cos  t 
 4 cos3  t   3cos  t 
cos  4t   cos  3t  cos  t   sin  3t  sin  t 
  4 cos3  t   3cos  t   cos  t    3sin  t   4sin 3  t   sin  t 
 4 cos 4  t   3cos 2  t   3sin 2  t   4sin 4  t 
 4 cos 4  t   3  4 1  cos 2  t  
2
 4 cos 4  t   3  4  8cos 2  t   4 cos 4  t 
 8cos 4  t   8cos 2  t   1