A Description and Method of Clairaut’s Equation

Clairaut’s equation is a differential equation in mathematics with the form y = x (dy/dx) + f(dy/dx), where f(dy/dx) is a function of just dy/dx. The equation is named after Alexis-Claude Clairaut, a French mathematician and physicist who invented it in the 18th century. He took part in an excursion to Lapland in 1736 with Pierre-Louis de Maupertuis with the goal of measuring a degree of the meridian, and upon his return, he published his treatise Theories de la figure de la terre (1743; “Theory of the Shape of the Earth”).He established the theorem, which links gravity at places on the surface of a rotating ellipsoid to compression and centrifugal force at the Equator, in this paper.

Definition:

Clairaut’s equation (or the Clairaut equation) is a differential equation of the form in mathematical analysis.

y(x) = x (dy/dx) + f(dy/dx)

where f can be differentiated indefinitely The Lagrange differential equation is a special case of it. It was first introduced in 1734 by the French mathematician Alexis Clairaut.

When one differentiates with regard to x, one gets Clairaut’s equation.

dy/dx = dy/dx + x(d²y/dx²) + f’(dy/dx).d²y/dx², so

[x + f’(dy/dx)] d²y/dx² = 0

Hence, either

d²y/ dx² = 0

Or

x + f’(dy/dx)

C = dy/dx for some constant C in the first example. When this is substituted into Clairaut’s equation, the family of straight line functions provided by.

y(x) = Cx + f(C),

Clairaut’s equation’s so-called generic solution.

The latter situation,

x + f’(dy/dx) = 0,

specifies a single solution The envelope of the graphs of the general solutions is y(x), the so-called singular solution. The singular solution is commonly written as (x(p), y(p)) in parametric notation, with p = dy/dx.

The single solution’s parametric description is of the form

x(t) = -f’(t)

y(t) = f(t) – tf’(t),

Where t is a parameter.

Extension:

A partial differential equation of the first order of the form

u = xux + yuy + f(ux,uy)

Clairaut’s equation is another name for it.

Symmetry of second derivatives:

The symmetry of second derivatives (also known as the equality of mixed partials) in mathematics relates to the ability of interchanging the order of taking partial derivatives of a function under specific conditions (see below)..

f(x1, x2, x3,………,xn).

With n variables The assumption that the second-order partial derivatives satisfy the identity is known as symmetry.

δ/δxi (δf/δxj) = δ/δxj (δf/δxi) 

So that they produce the function’s Hessian matrix, which is a n symmetric matrix Schwarz’s theorem, Clairaut’s theorem, and Young’s theorem are all names for the same thing.

It’s known as the Schwarz inerrability condition in the setting of partial differential equations.

Proof of Clairaut’s theorem using iterated integrals:

It is simple to establish the properties of repeated Riemann integrals of a continuous function F on a compact rectangle [a,b]*[c,d]. F’s uniform continuity indicates that the functions are all the same.

g(x) = ∫dc F(x,y)dy and

h(y) = ∫ba F(x,y)dx are continuous. It follows that

∫ba ∫dc F(x,y)dy dx = ∫dc ∫ba F(x,y)dx dy;

Furthermore, if F is positive, the iterated integral is immediately positive. The above equality is a straightforward application of Fubini’s theorem, requiring no measure theory. Titchmarsh (1939) proves it simply by utilizing Riemann approximation sums relating to rectangle subdivisions into smaller rectangles.

Assume f is a differentiable function on an open set U for which the mixed second partial derivatives fyx and fxy exist and are continuous to prove Clairaut’s theorem. Using the calculus fundamental theorem twice,

∫dc ∫ba fyx (x,y) dx dy = ∫dc fy(b,y) – fy(a,y) dy = f(b,d)-f(a,b) – f(b,c) + f(a,c).

∫ba ∫dc fxy (x,y) dy dx = ∫ba fx(x,d) – fx(x,c) dx = f(b,d) – f(a,d) – f(b,c) + f(a,c).

As a result, the two iterated integrals are equivalent. The second iterated integral, on the other hand, can be completed by first integrating over x and then over y because fxy(x,y) is continuous. The iterated integral of fyx fxy on [a,b]*[c,d] must, however, vanish. If, on the other hand, the iterated integral of a continuous function F vanishes for all rectangles, then F must be identically zero; otherwise, F or -F would be strictly positive at some point and hence by continuity on a rectangle, which is impossible. As a result, fyx and fxy must vanish in the same way, resulting in fyx = fxy everywhere.

ordinary differential equation:

In mathematics, an ordinary differential equation is an equation that connects a function f of one variable to its derivatives. (The adjective ordinary here refers to differential equations having only one variable, as opposed to partial differential equations, which involve numerous variables.)

The derivative of a function f, abbreviated f′ or df/dx, expresses the rate of change of the function at each point—that is, how quickly the function’s value grows or decreases as the value of the variable increases or decreases. The rate of change for the function f = axe + b (representing a straight line) is just its slope, written as f′ = a. The rate of change for other functions changes along the curve of the function, and differential calculus is used to define and calculate it precisely. The derivative of a function is, in general, another function, therefore the derivative of the derivative can be calculated as well, (f′)′ or simply f′′ or d2f/dx2, and is known as the second-order derivative of the original function. Higher-order derivatives can be defined in the same way.

A differential equation’s order is determined by the highest order derivative it includes. The power to which the highest order derivative is raised is the degree of a differential equation. The formula

(f’’’)² + (f’’’) + f = x

Is an example of a second-degree, third-order differential equation. If the function and all of its derivatives occur to the first power, and the coefficient of each derivative in the equation involves just the independent variable x, the equation is considered linear.

Some equations, such as f ′= x2, can be solved simply by remembering which function has a derivative that will satisfy the equation; however, in most cases, the solution is not obvious at first glance, and the subject of differential equations is partly concerned with classifying the various types of equations that can be solved using various techniques.

Conclusion:

Is Clairaut’s equation applied in any sector of science or engineering? Clairaut’s Equation is a first-order differential equation with the formula y=xy′+f(y′), where f(y’) is a non-linear function. The contrast between the general and singular solutions of an equation is highlighted by this equation.

Clairaut’s equation is a differential equation in mathematics with the form y = x (dy/dx) + f(dy/dx), where f(dy/dx) is a function of just dy/dx. The equation is named after Alexis-Claude Clairaut, a French mathematician and physicist who invented it in the 18th century.