Special solution of quadratic nonhomogeneous differential equation RT, how For example, y '' - 4Y '+ 4Y = f (x) Or y '' - (a + b) y '+ aby = f (x) (*) (a, B are constants) How to find the special solution of Only limited to the level of postgraduate entrance examination, is there any requirement for f (x) That is to say, if f (x) satisfies what kind of conditions, it can have a general special solution to the * formula

Special solution of quadratic nonhomogeneous differential equation RT, how For example, y '' - 4Y '+ 4Y = f (x) Or y '' - (a + b) y '+ aby = f (x) (*) (a, B are constants) How to find the special solution of Only limited to the level of postgraduate entrance examination, is there any requirement for f (x) That is to say, if f (x) satisfies what kind of conditions, it can have a general special solution to the * formula

You need a special solution. In fact, the special solution has something to do with your general solution. I'll summarize the general algorithm for you. It's my own review notes,
General solution of quadratic nonhomogeneous differential equation
The general formula is as follows: ay '' + by '+ CY = f (x)
The first step is to find the characteristic root
Let AR & sup2; + br + C = 0, then we get two values R1 and R2. (here it can be a complex number, for example, (β I) & sup2; = - β & sup2;)
Step 2:
If R1 ≠ R2, then y = C1 * e ^ (R1 * x) + C2 * e ^ (R2 * x)
If R1 = R2, then y = (C1 + c2x) * e ^ (R1 * x)
If r1,2 = α ± β I, then y = e ^ (α x) * (c1cos β x + c2sin β x)
Step 3:
The form of F (x) is e ^ (λ x) * P (x) (Note: P (x) is a polynomial about X, and λ is always 0)
Then y * = x ^ k * q (x) * e ^ (λ x) (Note: Q (x) is a polynomial of the same form as P (x), for example, P (x) is X & sup2; + 2x, then let Q (x) be ax & sup2; + BX + C, ABC are undetermined coefficients)
If λ is not the eigenvalue k = 0, y * = q (x) * e ^ (λ x)
If λ is a single K = 1, y * = x * q (x) * e ^ (λ x)
If λ is a double root, k = 2, y * = x & sup2; * q (x) * e ^ (λ x) (Note: the double root is solved above, R1 = R2 = λ)
The form of F (x) is e ^ (λ x) * P (x) cos β X or e ^ (λ x) * P (x) sin β X
If α + β I is not the eigenvalue, y * = e ^ λ x * q (x) (ACOS β x + bsin β x)
If α + β I is the characteristic root, y * = e ^ λ x * x * q (x) (ACOS β x + bsin β x) (Note: ab are undetermined coefficients)
Step 4: solve the coefficient of special solution
The y * ', y *', y * of the special solution are solved and brought back to the original equation
The final result is y = general solution + special solution
The coefficients C1 and C2 of the general solution are arbitrary constants
If you have any questions, you can ask me again. Take an example to illustrate the problem

First question:
The general solution of Y "- 6y '+ 9y = 0, and the special solution of Y | (x = 0) = 0 when y' | (x = 0) = 2
Second question:
The general solution of XY "- y '= 1
Can you give a detailed and easy to understand answer, I am a freshman, do not understand a lot of places

Let the second root of the equation be reduced to the characteristic equation: y = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) = x ^ 2 + (y) + (y) = y) = (y) = (y) = (y) = (y) = x ^ 2 + (y) = (y) = (y) = (y) = (y) = x ^ 2 + (Y

How to calculate 3.2/1.25,

=（3.2×8）/（1.25×8）
=25.6/10
=2.56

A number plus the reciprocal of seven sixths equals eleven ninths. Find this number

1/（11/9）-7/6
=9/11-7/6
=-23/66

Skillful calculation: (3 + 1) (3 ^ 2 + 1) (3 ^ 4 + 1) (3 ^ 8 + 1) mathematics of grade two
The answer on the Internet is yes, but I don't know if I can explain it

Multiply (3-1) by (3-1) and then divide (3-1) by (3-1) to get the square difference formula (3-1) (3 + 1) = 3 ^ 2-1 = (3-1) (3 + 1) (3 ^ 2 + 1) (3 ^ 4 + 1) (3 ^ 8 + 1) / (3-1) = (3 ^ 2-1) (3 ^ 2 + 1) (3 ^ 4 + 1) (3 ^ 8 + 1) / (3-1) = (3 ^ 4-1) (3 ^ 4 + 1) (3 ^ 8 + 1) / (3-1) = (3 ^ 16 -

56% of a number is equal to 20% of 18. Find the number

Let this number be X. & nbsp; & nbsp; & nbsp; 56x = 18 × 20% & nbsp; & nbsp; & nbsp; 56x = 3.6, 56x △ 56 = 3.6 △ 56, & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; X = 4.32. A: this number is 4.32

Ingenious calculation of 9.8-5.8 * 1.2 + 3.6 / 1.2

9.8-6.96+3
=5.84

Two thirds of a number is equal to one sixth of eighteen?

18×1／6÷2／3=4.5

3.8 * 9.9 emergency

=3.8*10-0.38=38-0.38=37.62
~If you agree with my answer, please click the "adopt as satisfactory answer" button in time
~The mobile questioner can click "satisfied" in the upper right corner of the client
~Your adoption is my driving force~

Why is (tanx-x) / x ^ 3 equal to x + 1 / 3x ^ 3 + O (x ^ 4) when the limit x tends to 0?

Replace TaNx with its expansion with the remainder of piano type, and then get the answer through four operations