CLAIRAUTS’ EQUATION
&
SINGULAR SOLUTION
GENERAL IDEA REGARDING CLAIRAUTS’EQUATION
Clairauts’ Equation is an ordinary first order differential equation. Its’ not solved with respect to its’ derivative.
Y=xy’+f(Y’), where f is continously differentiable. It is a particular case of the Langrange differential equation.
CLAIRAUT
The famous Clairaut’s Equation or Clairaut Equation was named after the French mathematician, astronomer and geo-physicist Alexis Clairaut who introduced it in 1734. Amongst his tons of achievements from confirmation of Newton’s Theory for the figure of Earth to tackling the gravitational three body problem, he worked out the mathematical result now known as Clairaut’s Theorem
DEFINITION
The Differential equation is of the form
Y=px+f(p)…(1) , p=dy/dx is called the Clairaut’s Equation
It’s complete primitive is
Y=cx+f(c)
(Replacing p by c in the given equation)
To solve Clairaut’s Equation,
It is differentiated with respect to x, yeilding
p = p+[x+ f’ (p) ]dp/dx
This gives either
dp/dx=0…(2)
Or
x+ f’(p)=0…(3)
When integration is done over (2) i.e dp/dx=0
We get, p=constant say c…(4)
From (1)and (4) we eliminate p and obtain the complete primitive in the form
Y=cx+f(c)
From (1) and (3) we obtain
x = -f’(p)…………….(5.1)
y= -f’(p) + f(p)……(5.2)
Since there are no arbitrary constant therefore it is not a general solution.
(5.1) and (5.2) together gives us the Singular Solution of (1)
Geometrically this represents a family of straight lines where singular solution is the envelope.
SINGULAR SOLUTION
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A singular solution
of an ordinary differential equation is a solution that is singular or one for which the initial value fails to have a unique solution at the same point on the solution.
by tangent we mean that there is a point x where ys (x) =yc (x)
And
where
is a solution in a family of solutions having c as a parameter.
This means the singular solution is a envelope of the family of solutions where the curve given by the singular solution and also at each point of the curve it is touched by some member of that family of lines.
Let’s take an example to illustrate the concept
Solve y= px +a/p
Taking (dy/dx)=p,
Differentiating with respect to x we get
p=p+ x(dp/dx) — (a/p2)dp/dx
Either, dp/dx =0…(1)
Or , x-a/p2 =0…(2)
From (1) ∫1 dp= ∫1.0. dx
We get p=constant say “c”
Putting p=c in the given equation,
y=cx+a/c…(3)
From (2) we get,
x=a/p2
hence y= c*(a/p2)+a/c
y=2a/p
Eliminating p we get y_2 =4ax
This is the singular solution.
I have used the following material as references:
*An Introduction To DIFFERENTIAL EQUATION by R K Ghosh and K C Maity
*https:en.m.wikipedia.org
This was a part of my tutorial/presentation in my undergraduate course. Hope this was beneficial to whoever comes across this and act as a helpful study guide!
Have A Good Day!!
