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

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. Separable variables

2. Homogeneous equations

3. Can be transformed into a homogeneous equation

4. First-order linear equation

5. Bernoulli’s equation

1.y＾n＝f（x）

2.y"=f(x,y')

3.y"=f(y,y')

y＂＋P（x）y＋＋q（x）y＝f（x）① y＂＋P（x）y＇＋q（x）y＝0 ②

1.Solution steps

2. Find the general solution Y for y"+py'+qy=0

3. Use the undetermined coefficient method to find the special solution y* of the non-homogeneous equation

y＾(3)＋py＂＋qy＋ry＝0

x＾ny＾（n）＋a1x＾（n-1）y＾（n-1）＋…＋an-1xy＋any＝f（x）