Analyzing 2nd Order Differential Equations with Resistive Components

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

Is there anyway to answer this question without solving the eqn and plotting the graph?
The function will not oscillate as there is -4y on the right side. So, the first option gets canceled.

Since there is a resistive part i.e. ## \frac{-dy} {dx} ##, the function has to decrease. But, when I plot the function, it first decreases to less than zero and then increases towards 0 as x tends to infinity.
And there is an extremum between o and 1.
So, how to say which option is corrrect, b or d ?

 

[DoItForYourself]

How do you know the function y(x)?

You have only one initial condition, but you need two (because you have 2nd order DE).

 

[Pushoam]

How do you know the function y(x)?

Because of the initial condition, I have only one arbitrary constant. I took arbitrary value of the constant e.g. 1, 2,500,2000,and so on and the graph had the same property.

Here, the question demands to know the property of the function on the basis of the given differential equation, before solving it. And I want to learn this skill.

 

[DoItForYourself]

Differentiate the solution function and set it equal to 0.

It appears difficult to find the right answer without solving the DE.

Also, have in mind that the arbitrary constant can be a negative number too.

 

[Pushoam]

Differentiate your function and set it equal to 0.

How does this help? This gets me to know the extremum part. I have done this. This doesn't say anything about part c and d.

I think I have taken the constant to be positive, so I got the min. to be between 0 and 1 and the function is initially decreasing to less than 0 and then reaches to 0. If I had taken the constant to be negative, then I would have got max. between o and 1 and the function would initially increase and then deccrease to 0. So, part c and d depends on the value of that arbitrary constant.
So, we can say only option b with certainty; and for this, too, we have to solve the eqn. Right?

 

[DoItForYourself]

Exactly. You cannot be sure if c or d is right because you do not know if the constant is negative or positive.

However, you know in which x the extremum of y(x) appears.

 

[Pushoam]

Yes. So, it is done.
Thank you.