九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2024年1月実施 解析学・微積分

Author

祭音Myyura (assisted by ChatGPT 5.5)

Description

(1) Find the maximum and the minimum of the function over the curve , and find the points which achieve them.

(2) Find the solution to the following differential equation defined for , where ,

(3) Find the radius of convergence of the series

where is a complex number.

Kai

(1)

We want to optimize

subject to

Use Lagrange multipliers:

where

Then

and

So

Solving this generalized eigenvalue problem gives

Because the constraint satisfies , the value of equals .

Case 1:

This gives

Using the constraint,

so

Thus the maximum value is

at the points

Case 2:

This gives

so

Substitute into the constraint:

Then

so

Hence

and

Thus the minimum value is

at the points

Therefore,

and

(2)

We solve

with initial condition

This is a first-order linear differential equation. The integrating factor is

Multiplying both sides by , we get

The left side becomes

So

Integrate both sides:

Using

we have

Divide by :

Use the initial condition :

Therefore,

for .

(3) Radius of convergence

Consider the power series

Let

Then the series becomes

We examine the general term

Using the root test,

This equals

For convergence, we need

Thus

Therefore, the radius of convergence is