To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides:

The integrals were solved using the following rules:

, 

The two constants of integration were combined to make a single constant.

Now, exponentiate both sides to isolate y, and use the properties of exponents to rearrange the integration constant:

(The exponential of the constant is another constant.)

Finally, we solve for the integration constant using the initial condition:

Our final answer is

To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides:

The integrals were solved using the following rules:

, 

The two constants of integration were combined to make a single one.

Now, we solve for y:

Because the problem statement said that y is negative - and y cannot be zero - our final answer is

To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides:

The integrals were solved using the following rules:

, 

The two constants of integration were combined to make a single one.

Now, we exponentiate both sides to solve for y:

Using the properties of exponents, we can rearrange the integration constant:

(The exponential of the constant is itself a constant.)

Using the given condition, we can solve for C:

Our final answer is

To begin with, the differential equation needs to be rearranged so that each variable is one side of the equation:

.  

Then, integrate each side of the rate law, bearing in mind that  will range from  to , and time will range from  to :

After integrating each side, the equation becomes:

.  

The left side has to be evaluated from  to , and the right side is evaluated from  to :

.  This becomes:

.  

Finally, rearranging gives:

This is a separable differential equation. The simplest way to approach this is to turn  into , and then by abusing the notation, "multiplying by dx" on both sides.

We then group all the y terms with dy, and all the x terms with dx.

Integrating both sides, we find 

Here, the first integral is found by using substitution of variables, setting . In addition, we have chosen to only put a +C on the second integral, as if we put it on both, we would just combine them in any case.

To solve for y, we multiply both sides by two and raise e to both sides to get rid of the natural logarithm.

(Note, C was multiplied by two, but it's still just an arbitrary constant. If you prefer, you may call the new C value .)

Now we drop our absolute value signs, and note that we can take out a factor of  and stick in front of the right hand side.

As  is just another arbitrary constant, we can relabel this as C, or  if you prefer. Solving for y gets us

Next, we plug in our initial condition to solve for C.

; 

Leaving us with a final equation of

Plugging in x = 4, we have a final answer,

To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides:

; 

The integrals were solved using the identical rules.

The two constants of integration are now combined to make a single one:

Now, exponentiate both sides of the equation to solve for y, and use the properties of exponents to rearrange C:

Finally, we solve for the integration constant using the given condition:

Our final answer is

First we notice that substituting 5 in for x will give us a 0 in the denominator.

So we simplify the equation by noticing the numerator is the difference of two squares.

Now we can substitute 5 in for x, and we arrive at our answer of 10.

Factor x-4 out of the numerator and simplify:

Evaluate the limit for x=4:

Although there is a discontinuity at x=4, the limit at x=4 is 10 because the function approaches ten from the left and right side.

Factor the numerator and simplify the expression.

Evaluate the function at x=2.

There is a discontinuity at x=2, but since it the limit as x approaches 2 from the right is equal to the limit as x approaches 2 from the left, the limit exists.

Factor the numerator to evaluate the limit:

Evaluate the limit:

There is a discontinuity at x=0 but the limit is equal to 8 because the limit from the right is equal to the limit from the left.