Separation of variables in ODE. Dear friends, today I will talk about the separation of variables in ordinary differential equations. This is one of the methods of solving first order ODE.
Separation of variables
As the name says, in this method, the variables are separated first.
Then both sides are integrated to get the solution to the equation.
Here I will show you two examples of ‘separation of variables’ method. Have a look!!
If interested, you can also check out more posts in first order ODE such as
- First-order homogeneous ODE
- Solve First-order ODE using transformations
- First-order linear ODE
- Bernoulli’s equations
Examples of separation of variables
Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.
So here is the first example.
Example 1
According to Stroud and Booth (2013)* “Find the general solution of ”
Solution
Here the given equation is
This equation does not have any term like . Here
can be real number except zero.
This means in this example it will be possible to separate and
variables.
In case if it would have any term like or
, then the ‘separation of variables’ method would not work.
Step 1
I will now try to separate and
components of this equation.
For this, I will divide the equation with to get
Now I will integrate both sides to the solution of the given equation.
Step 2
Thus the equation will look like
(1)
From the standard formula of integration, I already know that
Thus the left hand side of the equation (1) will be
(2)
Now I will work on the right hand side of the equation.
Suppose
From the standard rule of differentiation, I already know that if ,
In other words, I can also say that
Thus I can rewrite as
Now I will try to get the value of . Therefore
Here is the integration constant.
Now I substitute back in
to get
(3)
At the end, I will put the values of equations (2) and (3) in equation (1). So it becomes
So the conclusion is is the solution of the equation.
Now I will go to the second example.
Example 2
According to Stroud and Booth (2013)* “Solve given
when
”
Solution
Here the given differential equation is:
Like my first example, here also I will separate and
components.
Step 1
First of all, I will rewrite the equation as
Now I will divide the equation through out with to get
Next I will integrate both sides of the equation to get the solution.
Step 2
Thus the equation will look like
(4)
From the standard formula of integration, I already know that
Thus the left hand side of the equation (4) will be
(5)
For the right hand side of the equation (4), I already know that from the standard formula of integration. Thus it will be
(6)
Now I will put the values from (5) and (6) to (4) to get the solution of the given differential equation as
(7)
Well, this is the general solution of the equation.
Next I have to get the particular solution of the equation for
Step 3
Now I put and
in the equation (7) to get
Next I substitute this value of in the equation (7) to get the particular solution of the given equation as
Finally I can say that is the solution of the given equation for
Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!
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