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# Determine the interval of convergence of ∑ xn / n! and show that it satisfies y′ = x + y

Consider the function defined by the power series

Determine the interval of convergence for and show that it satisfies the differential equation

(We might notice that this is almost the power series expansion for the exponential function and deduce the interval of convergence and the differential equation from properties of the exponential that we already know. We can do it from scratch just as easily though.)

First, we apply the ratio test

Hence, the series converges for all . Next, we take a derivative

Then we have

Now, to compute the sum we can solve the given differential equation

This is a first order linear differential equation of the form with and . We also know that ; therefore, this equation has a unique solution satisfying the given initial condition which is given by

Where and

Therefore, we have

# Determine the interval of convergence of ∑ x2n / n! and show it satisfies a given differential equation

Consider the function defined by the power series

Determine the interval of convergence for and show that it satisfies the differential equation

First, to determine the interval of convergence we use the ratio test,

Hence, the series converges for all .

To show that it satisfies the given differential equation, we first take the derivative,

Then, it satisfies the given differential equation since

Then, since the given differential equation is a first-order linear differential equation of the form

with we know that the solutions are uniquely determined by the formula

Since we have the initial condition (by plugging in to the power series expansion for ) we have and the unique solution of this differential equation is

# Determine the interval of convergence of a given power series and show that it satisfies a given differential equation

Consider the function defined by the power series

Determine the interval of convergence for and show that it satisfies the differential equation

Find and .

First, to determine the interval of convergence for we use the ratio test,

Therefore, the series converges for all . Next, to show that it satisfies the given differential equation, we take the first two derivatives,

Then, we have the differential equation ,

But, for this equation to hold we must have (since there is no constant term in on the left) and we must also have since there the coefficient of on the left is 1 and the only possible term on the right is if . Using these values of and we verify that the given differential equation is satisfied since we have

Hence, we indeed have .

# Determine the interval of convergence of ∑xn / (n!)2 and show that it satisfies a given differential equation

Consider the function f(x) defined by the power series,

Determine the interval of convergence of f(x) and show that f(x) satisfies the differential equation

First, to determine the radius of convergence we use the ratio test,

Therefore, the series converges for all (or the radius of convergence is ). Next, to show that satisfies the given differential equation we take the first two derivatives,

Plugging this into the given differential equation we have

Hence, indeed satisfies the given differential equation.

# Determine the interval of convergence of ∑ x4n / (4n)! and show that it satisfies a given differential equation

Consider the function defined by the power series,

Determine the interval of convergence of and show that satisfies the differential equation

First, to determine the radius of convergence we use the ratio test

Therefore, converges for all (equivalently, ). Next, to show that satisfies the differential equation we take the first four derivatives,

But, reindexing this expression for the fourth derivative we have

Thus, satisfies the given differential equation.

# Use the method of undetermined coefficients to solve (1-x2)y′′ – 2xy′ + 6y = 0

Consider the differential equation

The solution to this differential equation has a power-series expansion

Using the method of undetermined coefficients obtain a recursion formula relating the terms to the terms . Give an explicit formula for for each and find the sum of the series.

First, we differentiate twice,

From the initial conditions and we have

Now, we plug the expressions for , and back into the given differential equation,

Then, we use the fact from above that and to get

Since this sum is equal to 0, we know that every coefficient of every power of must be equal to 0. First, we solve for and ,

Then, we establish the recursive relationship between and ,

for all . Then since we have for all (since for every odd integer the formula for has is multiplied by , but each of these will be 0). For the even terms we have for ,

This means all of the remaining even terms will be 0 as well. So we have

Hence,

# Prove that a given differential equation has a given particular solution

Let be defined by

for constants.

Assume and . Prove that the differential equation

has a particular solution of the form

and find expressions for and in terms of and .

Proof. Incomplete.

# Prove a given equation is a solution of a given differential equation

Let be defined by

for constants.

Let and . Using the previous exercise prove that the differential equation

has a particular solution given by

where

Proof. Incomplete.

# Prove existence of a complex valued solution to a given differential equation

Let be defined by

for constants.

Let and . If either or prove that the differential equation

has a solution

Express the value of in terms of , , , and .

Proof. Let with either or . Then we have

So,

But since or we have . Hence,

# Deduce properties of the solutions of a given differential equation

Let be a solution for all of the second order differential equation

Without attempting to solve the equation answer the following questions.

1. If and has an extremum at , prove that this extremum must be a minimum.
2. If has an extremum at 0 decide whether this extremum is a maximum or minimum and prove your assertion.
3. If the solution satisfies the conditions

find the minimum value for the constant such that

Incomplete.