All Calculus AB Resources
Example Questions
Example Question #1 : Find General And Particular Solutions Using Separation Of Variables
What is separation of variables?
Using algebra to rewrite a differential equation so that two different variables are on opposite sides
A fancy wording for factoring
Integrating with two variables
Taking the derivative of two variables
Using algebra to rewrite a differential equation so that two different variables are on opposite sides
When we are trying to integrate a differential equation, sometimes we have to use a method called separation of variables. This is because we need to integrate with respect to each variable but are unable to do so when they are on the same side. When we are able to get each variable on different sides with their respective differential (i.e. or ), then we can integrate each side with respect to each variable.
Example Question #1 : Find General And Particular Solutions Using Separation Of Variables
When would one need to use separation of variables?
We never actually have to use separation of variables it is just a shortcut
When we need to integrate a differential equation with two different variables
When we need to find the derivative of a differential equation
When we are solving a function with two different variables at a certain point
When we need to integrate a differential equation with two different variables
By using separation of variables we are able to integrate to solve the differential equation.
Example Question #2 : Find General And Particular Solutions Using Separation Of Variables
Use separation of variables to solve the following differential equation:
We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.
(divided by and multiplied by )
(just rearranging)
(Let )
( is just a constant so we rename it )
Example Question #3 : Find General And Particular Solutions Using Separation Of Variables
Use separation of variable to solve the following differential equation:
We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.
(multiplied by and multiplied by )
(Let )
( is just a constant so we will rename it )
Example Question #52 : Differential Equations
Use separation of variable to solve the following differential equations:
We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.
(multiplied by and )
(expansion)
(Let )
( is a constant so call it )
Example Question #2 : Find General And Particular Solutions Using Separation Of Variables
True or False: We can use separation of variables to solve a differential equation at a particular solution.
True
False
True
Just like integrating for a function of a single variable at a particular solution. We are also able to solve using separation of variables and integrating for our particular solution.
Example Question #6 : Find General And Particular Solutions Using Separation Of Variables
Find the general solution of the following differential equation at the point
.
We first must use separation of variables to solve the general equation, then we will be able to find the general solution.
(multiplied by )
(Let )
Now we plug in our particular solution to solve for our constant
And so our solution is
Example Question #4 : Find General And Particular Solutions Using Separation Of Variables
Find the general solution of the following differential equation at the point
.
(multiplying by and multiplying by )
(Let )
( is just a constant so rename it )
Now we plug in our point to solve for .
So our solution at this point is:
Example Question #5 : Find General And Particular Solutions Using Separation Of Variables
Find the particular solution for
using the point of the following differential equation.
We first must use separation of variables to solve the general equation, then we will be able to find the particular solution.
(multiplying by and )
(Let )
Now we plug in our initial condition that we were given
Now we will solve for when
Example Question #2 : Find General And Particular Solutions Using Separation Of Variables
What do we call a differential equation that can be solved by using separation of variables.
separable partials
they have no special name
separable equations
separable derivatives
separable equations
Separable equations are what we call differential equations that we are able to solve by using separation of variables.