In the previous post, we learnt about the basics of Clairaut's Equation. Now, let us see how easy it becomes to solve differential equations of the form
Let y'=p and re-write the equation
Now it becomes a Clairaut's Equation with
The GENERAL SOLUTION is simply
And the SINGULAR SOLUTION can be found very easily as
Now we can substitute the value of p in the original differential equation and obtain
Again see the graph
As we change the value of c in general solution, we get different solutions which here is a family of straight lines and notice how the singular solution remains tangent to every straight line we obtain through the general solution, that is, the singular solution is the envelope of general solution. Again, we solved a third degree first order differential equation so easily as it was a Clairaut Equation.
Let's put y'=p and simplify a bit
Let us substitute this value of p in the simplified differential equation
Now notice this has taken the form of Clairaut Equation, only the x and y terms have been replaced by x² and y². Can we apply the result of Clairaut's Equation in this equation too? Of course we can. Simply replace P with a constant c and we get our general solution.
Here is the graph of the general solution
How simple!
Now I will leave it up to you to find the SINGULAR SOLUTION. Can you tell if it exists? And if it does what does it represent? Remember to drop in your equations of the singular solution in the comments below.
Where p=y' using the following results
And the SINGULAR SOLUTION (if it exists)
We will also see what is an envelope of a curve.
Let us start by taking some examples. Before we solve the equation in the thumbnail, let us start by some direct and basic examples.
How will you solve?
Let us put y'=p and re-write the equation
Therefore, the general solution can be found very easily, simply replace p with a constant c, so the GENERAL SOLUTION bceomes
See how easily we found out the solution to our differential equation. No integration. Nothing. Simply by replacing p with c because it was in the form of a Clairaut's Equation. How easy!
Now let's find out the SINGULAR SOLUTION
Let us understand this through the graph of the general solution and the singular solution
Now if we see the graph, the general solutions are obtained by varying the value of the constant c. Here the general solution is a family of straight lines. Now the parabola you see is the singular solution. As it was said earlier, the singular solution is the envelope of the general solution, that is, the singular solution will be tangent to each and every curve we obtain through the general solution. Now it can be seen very clearly, that no matter what straight line we obtain by putting the value of c, it is always tangent to the parabola, our singular solution.
Now if we see the graph, the general solutions are obtained by varying the value of the constant c. Here the general solution is a family of straight lines. Now the parabola you see is the singular solution. As it was said earlier, the singular solution is the envelope of the general solution, that is, the singular solution will be tangent to each and every curve we obtain through the general solution. Now it can be seen very clearly, that no matter what straight line we obtain by putting the value of c, it is always tangent to the parabola, our singular solution.
Noticed how easily we solved a first order differential equation of degree 2?
Let's take a look at a similar example
How can we solve the differential equation
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgivm-vHMVhCEejtLjHsztyJLaBKqGbYa-7okl8fQBOr4NDNaYdx5hHoaB_7pgm0gXFTYiHBgRmmu5W98CiNPYQE71OW4HbbEae-ToWQP4mXKSarwIi2ySQSNl88FjgRL6m5PUetZwv760/s640/solution.gif)
Well, in these examples I took up, the equations were already in the form of Clairaut's Equation. It is also possible to reduce an equation in the form of Clairaut's Equation and solve it very easily.
Let us take a look at this.
How can we solve
Now it doesn't look like a Clairaut Equation
But is it possible to convert it into the form of Clairaut's Equation?
Let us try.
Let us make some substitutions
And
Differentiating
We get
And
Dividing equation 1 by equation 2 and simplifying, we get
Making the substitution and making further simplifications we get
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkarPekmKDA3fCpDRaSc1nAb_FNQVgqQPN3WABoOB89xF70NpdxpMZmiuUiMTrw6BZpLFubZBwMOKDHP14i13JuQBzawoU6M9KiUbSwGl5_3UQx3X7MAoP_QHYvL8cLmrzMtzUxrkcSLw/s1600/1594050165927275-3.png)
How simple!
Now I will leave it up to you to find the SINGULAR SOLUTION. Can you tell if it exists? And if it does what does it represent? Remember to drop in your equations of the singular solution in the comments below.
Till next time,
Thanks for reading
And
Happy Learning
Graphs created using Desmos
2 Comments
How Simple!
ReplyDeleteI know right! This is so fun
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