All AP Calculus AB Resources
Example Questions
Example Question #8 : Solving Separable Differential Equations And Using Them In Modeling
Find the solution to the differential equation
when .
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 #31 : Integrals
Differentiate .
While differentiating, multiply the exponent with the coefficient then subtract the exponent by one.
Example Question #31 : Integrals
If what is ?
Taking the derivative of gives you .
Taking the derivative of gives you .
Finally taking the derivative of gives you .
Therefore .
Example Question #9 : Solving Separable Differential Equations And Using Them In Modeling
Solve the following separable differential equation with initial condition .
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 :
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 .
Example Question #11 : Solving Separable Differential Equations And Using Them In Modeling
Solve the separable differential equation
given the condition
None of the other answers
To solve this equation, we must separate the variables such that terms containing x and y are on the same side as dx and dy, respectively:
Integrating both sides of the equation, we get
The integrals were found using the following rules:
,
After combining the constants of integration into a single C, exponentiating both sides, and using the properties of exponents to simplify, we get
To solve for C, we use the condition given:
Our final answer is
Example Question #12 : Solving Separable Differential Equations And Using Them In Modeling
is a function of . Solve for in this differential equation:
First, rewrite the expression on the right as the power of the radicand:
The expressions with can be separated from those with by multiplying both sides by :
Find the indefinite integral of both sides:
The expression on the right can be integrated using the Power Rule. On the right, use some -substitution, setting ; this makes and :
Apply some algebra to solve for :
Substitute back for , and apply some algebra:
Example Question #13 : Solving Separable Differential Equations And Using Them In Modeling
Solve the separable differential equation
where
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, keeping in mind rules for exponents which allow us to move the integration constant to the front:
To solve for the constant of integration, we use the condition given:
Our final answer is
Example Question #14 : Solving Separable Differential Equations And Using Them In Modeling
Solve the following separable differential equation:
given the condition that at
To solve the separable differential equation, we must separate x and y and their respective derivatives to either side of the equal sign:
Now, we integrate both sides of the equation:
The integrals were found using their identical rules.
Exponentiating both sides of the equation to solve for y - and keeping in mind the rules of exponents - we get
Now, we solve for the integration constant by using the condition given:
Our final answer is
Example Question #15 : Solving Separable Differential Equations And Using Them In Modeling
Solve the following separable differential equation:
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, where on the lefthand side, the following substitution is made:
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:
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