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MindMap Gallery differential equations

differential equations

Differential equations and first-order linear equations have only one arbitrary constant: y' p(x)y=q(x). Note: The absolute value can be omitted when calculating ∫p(x)dx. The second-order reducible differential equation: y''=f(y,y') type/y''=f(x,y') type/y''=f(y') type, can be reduced to the above two Use any type to find it, but generally use [missing y type].

Edited at 2023-08-20 00:32:58

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differential equations

first order equation

variable separable

y'=f(x)g(y), then ∫f(x)dx=∫dy/g(y) c

Note: Just mark a c

homogeneous type

mingle

Note: The meaning of homogeneity is to observe whether the powers of x and y are "aligned".

first order linear equation

There is only 1 arbitrary constant

y' p(x)y=q(x)

Note: The absolute value can be omitted when calculating ∫p(x)dx.

higher order equations

Second-order reducible differential equations

y''=f(y,y') type

y''=f(x,y') type

y''=f(y') type

It can be calculated as any of the above two types, but generally [missing y type] is used.

Higher Order Constant Coefficient Linear Differential Equations

The nature and structure of the solution <can be substituted>

Shaped like y'' p(x)y' q(x)y=0

If f(x)=0, that is, y'' p(x)y' q(x)y=0 is a homogeneous form

If f(x)≠0, that is, y'' p(x)y' q(x)y=0 is a non-homogeneous form

If it is a non-homogeneous special solution and y(x) is a homogeneous general solution, then the non-homogeneous general solution is: y is not (x)=y(x)

Feitong = Qitong Feite

If they are all homogeneous solutions

If the two are unrelated, that is, y1(x)/y2(x) is not equal to c, then y(x)=k1y1(x) k2y2(x) is a homogeneous general solution, where k1 and k2 are arbitrary constants (add to the problem superior.

If the correlation between the two is not determined, then y(x)=k1y1(x) k2y2(x) is a homogeneous solution, where k1k2 is an arbitrary constant

Qi Qi = Qi (linear combination)

If and are non-homogeneous special solutions

Then =- is a homogeneous solution

non-fei=qi

If and are respectively

Fei Fei = new non solution

If and y(x) are non-homogeneous special solutions and homogeneous solutions respectively, then

cy(x) is a homogeneous solution

c is a non-homogeneous special solution non-special = non-special homogeneous solution (a certain solution)

Find the general solution of the homogeneous equation y'' py' qy=0

has two roots

have a root

There is a pair of conjugate roots:

Special solutions of second-order non-homogeneous linear differential equations with constant coefficients

For example: y'' py' qy=f(x)

Special explanation: