Calculus Volume 3

# 7.4Series Solutions of Differential Equations

Calculus Volume 37.4 Series Solutions of Differential Equations

### Learning Objectives

• 7.4.1 Use power series to solve first-order and second-order differential equations.

In Introduction to Power Series, we studied how functions can be represented as power series, $y(x)=∑n=0∞anxn.y(x)=∑n=0∞anxn.$ We also saw that we can find series representations of the derivatives of such functions by differentiating the power series term by term. This gives $y′(x)=∑n=1∞nanxn−1y′(x)=∑n=1∞nanxn−1$ and $y″(x)=∑n=2∞n(n−1)anxn−2.y″(x)=∑n=2∞n(n−1)anxn−2.$ In some cases, these power series representations can be used to find solutions to differential equations.

Be aware that this subject is given only a very brief treatment in this text. Most introductory differential equations textbooks include an entire chapter on power series solutions. This text has only a single section on the topic, so several important issues are not addressed here, particularly issues related to existence of solutions. The examples and exercises in this section were chosen for which power solutions exist. However, it is not always the case that power solutions exist. Those of you interested in a more rigorous treatment of this topic should consult a differential equations text.

### Problem-Solving Strategy

#### Problem-Solving Strategy: Finding Power Series Solutions to Differential Equations

1. Assume the differential equation has a solution of the form $y(x)=∑n=0∞anxn.y(x)=∑n=0∞anxn.$
2. Differentiate the power series term by term to get $y′(x)=∑n=1∞nanxn−1y′(x)=∑n=1∞nanxn−1$ and $y″(x)=∑n=2∞n(n−1)anxn−2.y″(x)=∑n=2∞n(n−1)anxn−2.$
3. Substitute the power series expressions into the differential equation.
4. Re-index sums as necessary to combine terms and simplify the expression.
5. Equate coefficients of like powers of $xx$ to determine values for the coefficients $anan$ in the power series.
6. Substitute the coefficients back into the power series and write the solution.

### Example 7.25

#### Series Solutions to Differential Equations

Find a power series solution for the following differential equations.

1. $y″−y=0y″−y=0$
2. $(x2−1)y″+6xy′+4y=−4(x2−1)y″+6xy′+4y=−4$

### Checkpoint7.22

Find a power series solution for the following differential equations.

1. $y′+2xy=0y′+2xy=0$
2. $(x+1)y′=3y(x+1)y′=3y$

We close this section with a brief introduction to Bessel functions. Complete treatment of Bessel functions is well beyond the scope of this course, but we get a little taste of the topic here so we can see how series solutions to differential equations are used in real-world applications. The Bessel equation of order n is given by

$x2y″+xy′+(x2−n2)y=0.x2y″+xy′+(x2−n2)y=0.$

This equation arises in many physical applications, particularly those involving cylindrical coordinates, such as the vibration of a circular drum head and transient heating or cooling of a cylinder. In the next example, we find a power series solution to the Bessel equation of order 0.

### Example 7.26

#### Power Series Solution to the Bessel Equation

Find a power series solution to the Bessel equation of order 0 and graph the solution.

### Checkpoint7.23

Verify that the expression found in Example 7.26 is a solution to the Bessel equation of order 0.

### Section 7.4 Exercises

Find a power series solution for the following differential equations.

104.

$y ″ + 6 y ′ = 0 y ″ + 6 y ′ = 0$

105.

$5 y ″ + y ′ = 0 5 y ″ + y ′ = 0$

106.

$y ″ + 25 y = 0 y ″ + 25 y = 0$

107.

$y ″ − y = 0 y ″ − y = 0$

108.

$2 y ′ + y = 0 2 y ′ + y = 0$

109.

$y ′ − 2 x y = 0 y ′ − 2 x y = 0$

110.

$( x − 7 ) y ′ + 2 y = 0 ( x − 7 ) y ′ + 2 y = 0$

111.

$y ″ − x y ′ − y = 0 y ″ − x y ′ − y = 0$

112.

$( 1 + x 2 ) y ″ − 4 x y ′ + 6 y = 0 ( 1 + x 2 ) y ″ − 4 x y ′ + 6 y = 0$

113.

$x 2 y ″ − x y ′ − 3 y = 0 x 2 y ″ − x y ′ − 3 y = 0$

114.

$y ″ − 8 y ′ = 0 , y ( 0 ) = −2 , y ′ ( 0 ) = 10 y ″ − 8 y ′ = 0 , y ( 0 ) = −2 , y ′ ( 0 ) = 10$

115.

$y ″ − 2 x y = 0 , y ( 0 ) = 1 , y ′ ( 0 ) = −3 y ″ − 2 x y = 0 , y ( 0 ) = 1 , y ′ ( 0 ) = −3$

116.

The differential equation $x2y″+xy′+(x2−1)y=0x2y″+xy′+(x2−1)y=0$ is a Bessel equation of order 1. Use a power series of the form $y=∑n=0∞anxny=∑n=0∞anxn$ to find the solution. Do you know how you learn best?
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