Mathematical notes 12--Ordinary differential equations and discrete variables _ ordinary differential equations

Ordinary differential equations

Derivative that contains an unknown function, such as

The equation is a differential equation. In general, the equation that represents the relationship between the derivative of an unknown function, an unknown function, and an independent variable is called a differential equation. The unknown function is a unary function, called the ordinary differential equation, and the unknown function is called the partial differential equation of the multivariate function. This paper mainly introduces the ordinary differential equation.

The concept is often confusing, or look at the actual example:

The goal is to solve the relationship between X and Y. To convert an equation:

This is the final answer.

In fact, the process of solving ordinary differential is to use the knowledge of indefinite integral:

Separating variables

The separating variable is a method for solving the ordinary differential equation, which is suitable for the form of dy/dx = f (x) g (Y). Let's look at the following example:

In physics, it has a proprietary name called submerged operator. There is no need to dwell on the concept of physics, only to solve this equation mathematically. But this expression is not the same as the differential expression seen in the past, starting with the equation and converting it into a familiar form:

To solve the equation, you need to continue with the conversion:

That's the answer.

But the answer is only to solve the y>0 situation, y≤0 was not considered. A derivation can be used to verify whether the answer is the general solution:

Make a an arbitrary constant, convert the solution to Y=AE-X^2/2, and when a≠0, actually a=±a

The answer is general solution, the final answer is y=ae-x^2/2,a is arbitrary constant.

In fact, the answer is the normal distribution function, known as the Gaussian function, its prototype:

of which A,b,c∈r

The shape of the Gaussian function is like a hanging clock. A indicates the height of the curve, B is the offset of the centerline of the curve at the x-axis, the width of the C-half peak (the width of the function at half its peak).

When b=0,c=0,a=5, the image is as follows:

Y=AE-X^2/2 Example 1

The tangent of the curve intersects the straight line through the origin, and the tangent of the curve at the intersection is twice times the line slope, and the curve expression is obtained.

First, the above words are converted to equations, where the intersection point is (x,y), the curve is y=f (x), the slope of the curve tangent is y ', the line slope is y/x, and the following relationship is obtained:

by verifying to find the general solution, setting A=±a, a non-zero arbitrary constant, Y=AX2, verifies the solution:

The answer matches the original equation. The end result is y=ax2,a∈r,x≠0

When A=1, the curve y= x2,y ' =2x; the tangent slope of the (2,4) point is 4, the tangent is y=4x+b, the (2,4) is in the tangent, the 4=4x2+b,b=-4, and the tangent of the (2,4) point is y=4x-4. The following figure is a curve that satisfies the condition:

Y=ax2 is actually a family of curves:

Y=AX2 Example 2

Differential equation xdy/dx = (x2+x) (y2+1), find Y=f (x)

Here you need to review the derivation formula for trigonometric functions:

By the formula 15 above,

Validation, known trigonometric formula TAN2X+1=SEC2X

Example 3

d2y/dx2=6x, Y=f (x), y=f (x) has a horizontal tangent at (1,1) point.

The problem involves the second derivative and a limiting condition.

By limiting conditions to know:

Put (1,1) into the upper-type, 1 = 1–3 + C2,C2 = 3

Finally, y = x3–3x + 3 Summary The solution of ordinary differential equation by indefinite integral is a method of solving ordinary differential equation, which is suitable for the form of dy/dx = f (x) g (Y)