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differential equations
This figure summarizes the content of the differential equations chapter in Advanced Mathematics. The content is detailed and the steps are comprehensive. A differential equation containing the derivative or differential of an unknown function is called a differential equation. A differential equation in which the unknown function is a function of one variable is called an ordinary differential equation. The general form is f(x, y, y'...y(n))=0, and the standard form is y(n)=f(x, y, y'...y(n-1)).
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differential equations
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differential equations
basic concept
1. Definition: An equation containing the derivative or differential of an unknown function is called a differential equation, and a differential equation in which the unknown function is a function of one variable is called an ordinary differential equation. The general form is f(x, y, y'...y(n)) = 0, the standard form is y(n)=f(x, y, y'...y(n-1)).
2. Order: the highest order of the derivative or differential of the unknown function
3. Solution: function that satisfies the differential equation General solution: A solution that contains the same number of independent constants as the order of the equation Special solution: solution that does not contain arbitrary constants
4. Initial conditions: conditions for determining any constant in the general solution
theorem
1.y1(x), y2(x) are two second-order homogeneous (linearly independent) solutions, then any linear combination C1y1(x) C2y2(x) is also the (general) solution of this homogeneous differential equation .
2.y1(x), y2(x) are two second-order non-homogeneous linearly independent solutions. For any a b=1, ay1(x) by2(x) is also the solution of the non-homogeneous differential equation. For For any a b=0, ay1(x) by2(x) is the solution of the homogeneous equation corresponding to the equation. In the same way, three solutions are three coefficients.
3. If C1y1(x) C2y2(x) is the general solution of the second-order homogeneous equation, and y*(x) is the special solution of the second-order non-homogeneous equation, then C1y1(x) C2y2(x) y*(x) is not Homogeneous and convenient general solution. 【Feitong=Qitong Feite】
4. If y*1(x) is a special solution of the second-order non-homogeneous equation 1, and y*2(x) is a special solution of the second-order non-homogeneous equation 2, then y*1(x) y*2( x) is a special solution of the second-order non-homogeneous equation 1 and equation 2. [The principle of superposition of solutions]
5.y1, y2, and y3 are three non-homogeneous linearly independent solutions. The general solution is y=C1 (y2-y1) C2 (y3-y2) y3 [subtract any two, and finally add any one solution]
6. The difference between two non-homogeneous special solutions is a homogeneous special solution (used in 5)
1. First-order differential equation
1. Separable variables
y'=f(x)·g(y)
Solution: ∫dy/g(y)=∫f(x)dx
2. Homogeneous equations
y'=f(y/x)
solution: ①Let u=y/x, y＇＝u＋xu＇ ②So u+xu′=f(u) ③∫du／［f（u）-u]=∫dx／x＝lnIxl＋C ④u restores u to y/x
3. Can be transformed into a homogeneous equation
4. First-order linear equation
y＇＋P（x）y＝Q（x）
solution: 1. Homogeneous general solution: y=Ce＾(-∫P(x)dx) 2. Non-homogeneous general solution: y＝e＾（-∫P（x）dx）[∫Q（x）e＾（∫P（x）dx）＋C]
5. Bernoulli’s equation
y＋P（x）y＝Q（x）y＾n（n≠0，1）
solution: ①Let z=y＾(1-n) ②Then z＋(1-n)P(x)z＝(1-n)Q(x) ③According to the 4 solution method
2. Reducible higher-order differential equations
1.y＾n＝f（x）
Solution: n times of integration
2.y"=f(x,y')
Solution: Let y'=p, turn into p'=f (x, p)
3.y"=f(y,y')
solution: ①Let y'=p, y"=pdp/dy ② Change to pdp/dy=f(y,p) ③Separate the variables after deriving the derivative of y...
3. Structure of solutions to linear differential equations
y＂＋P（x）y＋＋q（x）y＝f（x）① y＂＋P（x）y＇＋q（x）y＝0 ②
1. If y1(x) and y2(x) are two linearly independent solutions of ②, then y=C1y1 C2y2 is the general solution of ②
2. If y* is a solution of ①, and y=C1y1 C2y2 is the general solution of ②, then y=y* C1y1 C2y2 is the general solution of ①
3.y1 is the solution of ①1, y2 is the solution of ①2, then y1 y2 is the solution of ①1 ①2
4. Solution to second-order constant coefficient linear differential equations
1.Solution steps
① Find the general solution Y of the corresponding homogeneous equation
② Find the special solution y* of the non-homogeneous equation
③Write the general solution y=Y y*
④Find the special solution according to the definite solution conditions
2. Find the general solution Y for y"+py'+qy=0
1. Write the characteristic equation r＾2＋pr＋q＝0 and find r1, r2
2. Write the general solution of the equation
①When r1≠r2, Y＝C1e＾（r1x）＋C2e＾（r2x）
②When r1=r2, Y=(C1+C2x)e＾(r1x)
③When r1, 2=α±阝i, Y=e＾(αx)〈C1cos阝x＋C2sin阝x〉
3. Use the undetermined coefficient method to find the special solution y* of the non-homogeneous equation
y＂＋py＋qy＝
type
Pm(x)e＾(αx)
e＼（αx）〈Pl（x）cos阝x＋Pm（x）sin阝x〉
1. Set y*= according to the equation
x＾kQm（x）e＾（αx）
x＾ke＾（αx）〈Rn（x）cos阝x＋Tn（x）sin阝x〉
Among them, n=max(l,m), Qm(x) is a general polynomial of degree m of x, if Pm(x)=x^2, then Qm(x)=ax^2 bx c, Rn(x), Tn(x) is a general polynomial of degree n of x
2. The choice of k is divided into the following situations:
①When α or α 靝i is not a characteristic root, k=0
②When α or α 阝i is a single root of the characteristic equation, k=1
③When α is a multiple root of the characteristic equation, k=2
f(x)=Pn(x) Pn(x) is a polynomial of degree n
①0 is not a characteristic root: y*=Hn(x)
②0 is the characteristic single root: y*=xHn(x)
③0 is the characteristic multiple root: y*=x^2Hn(x)
f(x)=Pn(x)e^（αx）
①α is not a characteristic root: y*=Hn(x)e^(αx)
②α is a characteristic single root: y*=xHn(x)e^(αx)
③α is a characteristic multiple root: y*=x^2Hn(x)e^(αx)
f(x)=e＾（αx）〈Pn（x）sin阝x＋Qm（x）cos阝x〉
α靝i is not a characteristic root: y*=e＾(αx)〈Rl(x)cos阝x＋Sl(x)sin阝x〉
α靝i is the characteristic root: y*=xe＾(αx)〈Rl(x)cos阝x＋Sl(x)sin阝x〉
l=max(n,m)
5. Solution to third-order homogeneous linear differential equations with constant coefficients
y＾(3)＋py＂＋qy＋ry＝0
1. Write the characteristic equation into ＾(3)＋p into ＾(2)+q into ＋r=0
Write the general solution to the equation
①When input 1≠enter 2≠enter 3 (single real root), Y＝C1e＾(enter 1x)+C2e＾(enter 2x)+C3e＾(enter 3x)
② When entering 1 = entering 2 ≠ entering 3 (double real root), Y = [C1 + C2x] e＾ (entering 1x) + C3e＾ (entering 3x)
③ When entering 1 = entering 2 = entering 3 (triple real root), Y = [C1＋C2＋C3x^2]e＾ (entering 1x)
④ Enter 1, 2=α±阝i, enter 3∈ single real root, Y＝e＾(αx)〈C1cos阝x＋C2sin阝x〉+C3e＼(enter 3x)
6. Euler’s equation
x＾ny＾（n）＋a1x＾（n-1）y＾（n-1）＋…＋an-1xy＋any＝f（x）
1. Let x＝e＾t d／dt＝D, d＾2／dt＾2＝D＾2,…, d＾n／dt＾n＝D＾n 2. Then xy＇＝Dy＝dy／dt x＾ny＾（n）＝D（D-1）…（D-n＋1）y 3. Substituting into the original equation, it can be transformed into a high-order constant coefficient linear differential equation.