This is separable differential equation. The first step is to get all of the x components of the function on one side and the y components on the other. Once that is done, integrate each side as shown:

To evaluate the integral, use the inverse power rule

.

Integrating the function should result in the answer below:

Given the linear, homogenous differential equation 

The general solution is given by 

Since , 

The specific solution is therefore

To find the derivative of this function we must use the product rule.

It states that the derivative of  is .

We must also recognize that the derivative of sine is cosine and the derivative of  is .

The derivative of this function is then

.

To find the derivative of of this function we must use the quotient rule.

It states that the derivative of 

 is .

We must also use the rules that the derivative of  is  and that the derivative of  is .

Thus the derivative of the function is

.

To calculate this derivative we must use the chain rule.

The chain rule states that the derivative of  is .

We must also understand that the derivative of sine is cosine, the derivative of  is , and the derivative of  is .

 so we must take the derivative of the inner function which is .

We must also use the chain rule for this.

The derivative of  is .

The derivative of the outer function is .

Thus the final answer is

.

To solve this equation, notice that the second derivative of a function  is equal to the negative of the original function. 

The only elementary functions that satisfy this rule are the sine and cosine functions.

Therefore, possible solutions to this differential equation is  and .

To find the general solution for the differential equation, we must bring the y and dy terms to the same side, and the x and dx terms to the same side:

Now, integrate on both sides:

We used the following rules for integration:

, 

Note that we only have one constant of integration, C, because the one from the left side of the equation (from the y integration) was combined with the one on the right side.

Now, exponentiate both sides of the equation to finish:

Here we use the guess and check method to see if we can find a solution to the differential equation. We can find the first derivatives of each of the answer choices with respect to  and then perform the operations on the derivative indicated to see which of the choices satisfies the differential equation. Taking these derivates involves our using the product rule and the chain rule, both given below. 

  ,  

Here is a list of each of the answer choices, their derivatives and then the first few steps of the operations given in the differential equation. 

From here you can clearly see that one answer will satisfy the differential equation.

To solve the differential equation we can integrate both sides with respect to .

The left hand side just becomes  as 

(Note we can save the  and just include that on one side of the equation. The right hand side becomes 

as we are integrating a constant, and the integral of any constant, , is just   by 

. 

Now that we have the expression 

, we can use our initial condition to solve for the constant of integration, .

Since we have , then .

Therefore,  and our specific solution is .

The differential equation can be seperated, so that y and dy and x and dx are on the same sides:

Now, integrate both sides:

We used the following integration rules:

, 

Note that the C's combined to make one C.

Finally, exponentiate both sides to get our final answer: