Let N be a positive integer. Find all real numbers a such that the differential equation d2ydx2−4adydx+3y=0 has a nontrivial solution satisfying the conditions y(0)=0 and y(2Nπ)=0

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meli199939f

2022-03-23

Let N be a positive integer. Find all real numbers a such that the differential equation

\frac{d^{2} y}{d x^{2}} - 4 a \frac{d y}{d x} + 3 y = 0

has a nontrivial solution satisfying the conditions

y (0) = 0

and

y (2 N π) = 0

kanonickiuoeh

Beginner2022-03-24Added 8 answers

Step 1
You found that

e^{(m_{2} - m_{1}) 2 N π} = 1

If we go one step further,

e^{4 \sqrt{4 a^{2} - 3} N π} = 1

You need to consider complex solutions as well:

e^{2 π k i} = 1

for any

k \in Z

. Then

k = 0

is the solution you already found.So you have

4 a^{2} - 3 \leq 0

and

2 N \sqrt{3 - 4 a^{2}} \in Z

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