Equations of elliptical partial differential equations, differential equations of ellipses

Science Encyclopedia: Partial Differential Equations.

An ellipse is the trajectory of a point on a plane where the sum of the distances to two fixed points is constant.

It can also be defined as the trajectory of a point where the ratio of the distance to a fixed point and the distance between a fixed line is a constant value less than 1.

It is a conic curve.

A kind of cross-section, that is, the section of the conic and the plane.

An ellipse can be written on the equation as:

x²/a²+y²/b²=1

It also has some other forms of expression like parametric equations.

representation and so on.

Solution: The differential equation with respect to y is f'(0,y) f(0,y)=coty, there is f'(0,y) f(0,y)=cosy siny, and both sides are integrated at the same time with ln|f(0，y)|=ln|siny|

ln|c|(c is any non-zero constant), get:

f(0,y)=csiny, and when x=0, c(x)=c differential equation f(x,y) x=-f(x,y), at this time.

Think of y as a constant, and partial differential equations can be seen as such.

Ordinary differential equation df(x,y) dx=-f(x,y), yes.

df(x，y)/f(x，y)=-dx，ln|f(x，y)|=x+ln|c|(where c is the equation for y, and c ≠0), we get: f(x,y)=c(y)e (-x)).

Then c(y)e (-x)=c(x)siny, and the equation z=f(x,y) is.

z=siny×e^(-x)

An equation that contains a partial derivative (or partial differential) of an unknown function. The highest order of the partial derivative of an unknown function that appears in an equation is called the order of the equation.

The discipline of calculus equations arose in the eighteenth century, when Euler first proposed the second-order equation of string vibration in his work, and soon after, the French mathematician d'Alembert also proposed special partial differential equations in his work "On Dynamics". However, these writings did not attract much attention at the time.

In 1746, d'Alembert, in his Study of the Curves Formed When a Tensioned String Vibrates, proposed to prove that an infinite variety of curves that differ from the sinusoidal curve are modes of vibration. In this way, the study of chord vibrations led to the creation of the discipline of partial differential equations.

Mathematical Applications

Mathematically, the initial and boundary conditions are called solution conditions.

The partial differential equation itself expresses the commonality of the same type of physical phenomenon and serves as the basis for solving problems. However, the solution conditions reflect the personality of the specific problem, and it presents the specific situation of the problem. When the equation and the solution condition are combined, it is called a solution problem.

To find the solution of a partial differential equation, you can first find its general solution, and then use the solution conditions to determine the function. However, in general, it is not easy to find the general solution in practice, and it is even more difficult to determine the function with the definite solution condition.

Partial differential equations are an important branch of mathematics, which is a mathematical model for describing natural phenomena and physical phenomena. Partial differential equations are often used to describe the variation of some variables with time, space and other factors. They can be used to solve many important practical problems, such as problems in fluid mechanics, electromagnetism, heat conduction, quantum mechanics, and more.

Partial differential equations can be divided into several types, including:

1.Elliptic partial differential equations: used to describe steady-state problems, such as electrostatic fields, magnetostatic fields, etc.

2.Parabolic partial differential equations: used to describe problems such as heat conduction, diffusion, fluctuations, etc.

3.Hyperbolic partial differential equations: used to describe fluctuations, **, etc.

Methods for solving partial differential equations include the separation variable method, the transformation method, and the numerical method. In practical applications, the solution of partial differential equations often requires a combination of numerical methods and computer simulations.