It is proved that the second order linear ordinary differential equation has two linearly independent solutions The equation is as follows: y '' + P (x) * y '+ Q (x) * y = 0; It is proved that the differential equation must have two linearly independent solutions; How to prove it? Why must it be two? And linearly independent?

It is proved that the second order linear ordinary differential equation has two linearly independent solutions The equation is as follows: y '' + P (x) * y '+ Q (x) * y = 0; It is proved that the differential equation must have two linearly independent solutions; How to prove it? Why must it be two? And linearly independent?

A general n-order linear ordinary differential equation must have n linearly independent solutions
It takes quite a long time to prove. For the case of level 2, we can consider the following points
1) If the equation has two linearly independent solutions, its linear combination must also be the solution of the original equation (this is the superposition principle)
2) If the equation has two linearly independent solutions, the corresponding lungsky determinant can be obtained by substituting the two solutions into the original equation. At this time, the lungsky determinant must not be zero in the corresponding interval. It is known from linear algebra that two linearly independent solutions can form the general solution of the original equation; at the same time, it is known that one solution cannot express the general solution
3) If the equation has three linearly independent solutions, then two linearly independent solutions can be obtained by subtracting each other. According to 2), we can know that the three linearly independent solutions are contradictory
Finally, we summarize the above, which is the general solution structure theorem (LZ's topic is only a small part of the theorem)

Why are linear ordinary differential equations with constant coefficients homogeneous solutions and special solutions

1: By substituting the homogeneous solution and special solution into the linear ordinary differential equation with constant coefficients, the solution is determined
2: The solution has any homogeneous constant
Therefore:
Homogeneous solution plus special solution is the general solution

How to judge linear space
Determine whether the following sets are linear spaces
(1){（x,y,z）/ x+2y+z=0};
(2){（x,y,z）/ z=x+y};
(3){（x,y,z）/ z=xy};
I don't understand that. Can we give examples or use formula derivation,

1 and 2 are linear spaces, 3 is not
Linear space is simple, linear space is such a set, in which any two elements can be added to form another element in the set, so 1 and 2 conform to the definition, which is linear space, while 3 is the opportunity of two elements, which does not conform to the definition, so it is not linear space

Judgment of linear space
All positive real numbers, addition and multiplication are defined as:
a○b=ab
k°a=a^k,
Does the set constitute a linear space over a real number field for the linear operation it refers to?
PS: if it's a linear space, do we have to have 0 and 1 in the set?

Yes
Because addition has zero elements, that is, there is element 0, so that x + 0 = 0 + x = X
The number 1 makes 1 * x = X

What is linear decision making

It also allows me to see you, vegetable
Decision criterion method
Decision criteria are the principles that decision-makers should follow in the whole process of decision-making, including the thinking mode of decision-making, decision-making organization, and the principle requirements of making alternatives. According to the "economic man" model, people always adopt the "optimization principle" when evaluating and selecting various feasible schemes, that is, people always hope to compare various feasible schemes, For this kind of decision-making criteria, it needs to meet the following conditions: (1) before making a decision, it needs to comprehensively look for alternative behaviors; (2) it needs to examine all the complex consequences caused by each possible choice; (3) it needs to have a set of value system as the selection criteria for selecting one of the alternative behaviors, When using the principle of optimization to make decisions, decision makers must find all possible decision-making schemes before making decisions. At the same time, they must be able to estimate the results of the implementation of each scheme in advance. Finally, there must be a unified value criterion, which can rank the results of various schemes continuously and consistently

It is proved that the function f (x, y) = sqrt (lxyl) is continuous at (0,0), and the partial derivative exists, but it is not differentiable at (0,0)

Root (| XY |)

Find the derivative (or differential) of the following function:
(1) Y = ln (3x ^ 2 + e ^ 2), find dy / DX;
(2) Y = ∫ 0 ^ x cost ^ 2 DT, find dy / DX;
(3) Y = xsinx, find y '';
(4) Y = (2xsinx + 3) / x, find dy;
(5) 3-x ^ 2 = 2Y ^ 3-y, find dy / DX;
(6) Y = (SiNx) ^ LNX, find dy / DX
Please help me, my mathematics is really bad, need to test, so need to have detailed steps and answers, the answer satisfaction continue to add reward more than 30, here first thank you, very urgent

(1) Y = ln (3x ^ 2 + e ^ 2), find dy / DX;
dy/dx=6x/[3x²+e²]
(2) Y = ∫ 0 ^ x cost ^ 2 DT, find dy / DX;
=∫(1/2)(1+cos2t)
=(1/2)t+(1/4)sin2t+C [0,x]
=x/2+(1/4)sin2x
dy/dx=1/2+(1/2)cos(2x)
(3) Y = xsinx, find y '';
y'=sinx+xcosx
y''=cosx+cosx-xsinx
=2cosx-xsinx
(4) Y = (2xsinx + 3) / x, find dy;
=2sinx+3/x
dy=2cosx-3/x² dx
(5) 3-x ^ 2 = 2Y ^ 3-y, find dy / DX;
Derivation on both sides
-2xdx=6y²-1dy
dy/dx=-2x/(6y²-1)
(6) Y = (SiNx) ^ LNX, find dy / DX
Logarithm
lny=lnx*lnsinx
Derivation
y'/y=(1/x)lnsinx+(1/tanx)lnx
y'=[(1/x)lnsinx+(1/tanx)lnx]*[(sinx)^lnx]
dy/dx=[(1/x)ln(sinx)+(1/tanx)lnx](sinx)^lnx

What are the conditions of differential function in Higher Mathematics?

What kind of function do you mean? If the function of two variables requires that the function be continuous at the change point, if the function of multiple variables requires that each of the partial derivatives at the change point exist

A method to find the second order partial derivative of implicit function of multivariate function
For example: four second order partial derivatives of Z ^ 3-2xz + y = 0

There are only three second-order partial derivatives, and there are only three second-order partial derivatives, which are only three second-order partial derivatives, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\; &# 178; Z / (&; Y &; x) is equivalent to the partial order

How to calculate the derivative dy / DX of X / y = ln (XY), I don't know how to connect X and Y together as the differential of a term
The derivative dy / DX of X / y = ln (XY) and the derivative dy / DX of 2x ^ 2y-xy ^ 2 + y ^ 3 = 0

I can do it, this is an implicit number, both sides of X derivation, (Y-Y *) / y ^ 2 (use it to express y to x derivation) = [1 / (XY)] × (XY * + y), and then y * = (xy-y ^ 2) / X (1 + x), similarly, to x derivation, the solution, y * = y ^ 2 / (8yx ^ (2y-1) - 2XY + 3Y ^ 2), if you don't agree, please discuss