# AP Calculus AB : Solving separable differential equations and using them in modeling

## Example Questions

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### Example Question #1 : Solving Separable Differential Equations And Using Them In Modeling

Find (dy/dx).

sin(xy) = x + cos(y)

dy/dx = (xcos(xy) + sin(y))/(1 – ycos(xy))

None of the above

dy/dx = (cos(xy) + sin(y))/(1 – cos(xy))

dy/dx = (1 – cos(xy))/(cos(xy) + sin(y))

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

Explanation:

The first step of the problem is to differentiate with respect to (dy/dx):

cos(xy)[(x)(dy/dx) + y(1)] = 1 – sin(y)(dy/dx)

*Note: When differentiating cos(xy) remember to use the product rule. (xy' + x'y)

Step 2: Clean the differentiated problem up

cos(xy)(x)(dy/dx) + cos(xy)y = 1 – sin(y)(dy/dx)

cos(xy)(x)(dy/dx) + sin(y)(dy/dx) = 1 – cos(xy)y

Step 3: Solve for (dy/dx)

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

### Example Question #1 : Applications Of Antidifferentiation

Find the equation of the normal line at  on the graph .

Explanation:

Now plug in .

now we know 6 is the slope for the tangent line. However, we aren't looking for the slope of the tangent line. The slope of the normal line is the negative reciprocal of the tangent's slope; meaning the slope of the normal is . Now find the equation of the normal line.

### Example Question #3 : Solving Separable Differential Equations And Using Them In Modeling

What is the derivative of

Explanation:

Use the quotient rule.

Find  if

Explanation:

### Example Question #5 : Solving Separable Differential Equations And Using Them In Modeling

Find the derivative:

Explanation:

To find the derivative, multiply the exponent by the coefficent in front of the x term and then decrease the exponent by 1:

### Example Question #6 : Solving Separable Differential Equations And Using Them In Modeling

Find the solution to the equation at with initial condition .

Explanation:

First, we need to solve the differential equation of .

, where is a constant

, where is a constant

To find , use the initial condition, , and solve:

Therefore, .

Finally, at , .

### Example Question #7 : Solving Separable Differential Equations And Using Them In Modeling

Solve the differential equation:

Note that  is on the curve.

Explanation:

In order to solve differential equations, you must separate the variables first.

Since point  is on the curve, .

To get rid of the log, raise every term to the power of e:

### Example Question #8 : Solving Separable Differential Equations And Using Them In Modeling

Find the solution to the differential equation

when .

Explanation:

First, separate the variables of the original differential equation:

.

Then, take the antiderivative of both sides, which gives

Use the given condition , plugging in

and , to solve for .  This gives , so the correct answer is

.

### Example Question #9 : Solving Separable Differential Equations And Using Them In Modeling

Solve the following separable differential equation  with initial condition .

Explanation:

We proceed as follows

. Start

. Rewrite  as .

. Multiply both sides by , and divide both sides by .

. Integrate both sides. Do not forget the  on one of the sides.

Substitute the initial condition .

.

. Solve for .

. Exponentiate both sides .

. Rule of exponents.

### Example Question #10 : Solving Separable Differential Equations And Using Them In Modeling

Solve the separable, first-order differential equation for

Explanation:

Solve the separable, first-order differential equation for

First collect all the terms with the derivative  to one side of the equation.

Important Conceptual Note: often in texts on differential equations differentials often appear to have been rearranged algebraically as if  is a "fraction," making it appear as if we "multiplied both sides" by  to get:  . This is not the case. The derivative is a limit by definition and, when the limit exists, can take on any real number which includes irrational numbers i.e. numbers which cannot be written as a ratio of two integers.

For instance, we cannot represent  as a ratio, but some functions may have a derivative at a point such that the derivative is equal to , or a funciton may simply have an irrational number like  as a derivative. For instance, if   we write the derivative . Claiming that  and  are representative of a "numerator" and "denominator" respectively, we would essentially be claiming to have found a way to write an irrational number, such as  as a ratio, which is preposterous. The expression  is simply notation.

Here is what we are really doing.

Note that the constants of integration can just be combined into one constant by defining  .

Solve for :

Applying the initial condition:

Here we have two possible solutions. However, because of the initial condition, we can easily rule out the negative solution.  must be equal to positive

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