Second Order Derivative

To simplify the given equation using the substitution , we follow these steps:

Step 1: Express trigonometric functions in terms of :
Let , so that

Using known identities:

Step 2: Chain Rule — Convert derivatives with respect to into derivatives with respect to :

Using , we differentiate:

Now express and in terms of , then substitute back into the differential equation. After full simplification:

The resulting equation is:

Correct Answer: (C)

The given equation simplifies to:



Step 1: Analyze the structure
This is a second-order differential equation in terms of and . For the equation to be meaningful, we consider when it leads to real and distinct solutions for .

Step 2: Consider the nature of the solution
This differential equation implies that the second derivative of with respect to is constant. Therefore, integrating twice, we get:


Step 3: Interpret the problem context
Usually, such questions ask for the number of values of that satisfy a certain condition. If an earlier condition or context (from a previous step) implies that a specific value of is known (e.g., 0), then:

Set:

Hence, there are two distinct values of that satisfy this condition: and

Final Answer: (B) Two distinct values of

Step 1: Define x and y.

We are given and .

Step 2: Calculate y1 (dy/dx).

First, we find and :

Now, we can find using the chain rule:

Step 3: Calculate y2 (d^2y/dx^2).

We need to find . We use the chain rule again:

First, let's find :

Now, we can find :

Step 4: Substitute into the given equation.

We are given . Substitute the expressions for and :

Since , we can simplify:

Step 5: Solve for α.

For the equation to hold true for all , we must have:

Final Answer

Therefore, .