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:

To solve the seperable differential equation, we must put the x and dx and y and dy on the same sides:

Now, integrate both sides:

The integrations were performed using the following rules:

Finally, solve for y:

Note that the Cs combined to make one constant of integration.

The solution for the separable differential equation can be found by first separating x and dx, y and dy:

Now, integrate both sides:

, 

The following rules were used for integration:

, 

Finally, solve for y:

Note that the Cs combined to make one constant of integration.

To find this derivative, use of both the product rule and quotient rule for derivatives will be necessary. The latter states:

For our function

The  derivative is simply 

For the  derivative, the product rule will be useful:

Where 

And the derivative is:

Putting all of this together, the derivative of f(x) is:

To solve the differential equation, we must move the x and y terms with dx and dy, respectively:

Now we can integrate:

using the following rules:

, 

To finish, write the equation in terms of y alone:

This can be easily separated into two derivatives added together:

The second function is easy: the derivative of any constant is 0. But for the first, we must use the chain rule.

Recall:

Our outside function is  and our inside function is .

So the chain rule tells us we must take the derivative of  and plug  into that function.

The derivative of  is , so we have

.

Now all we have to do is find the derivative of , which we know is .

So our final answer is 

.

By the power rule, we know that

, where  are constants and  is a variable.

In our case,

, where  is a constant.