General and special solutions of differential equations XY '= 2Y why is y = 5x ^ 2 a special solution of the differential equation? Isn't the definition of a special solution a solution without any constant? Why isn't 5 an arbitrary constant?

General and special solutions of differential equations XY '= 2Y why is y = 5x ^ 2 a special solution of the differential equation? Isn't the definition of a special solution a solution without any constant? Why isn't 5 an arbitrary constant?

Any constant is C
5 is a specific constant
That is, if your solution is CX ^ 2 (the general solution of Y '= 2x * y), it holds for any constant C, it is called the general solution
5x ^ 2 has only a fixed number, not a general solution

Find the general solution or special solution of the following differential equation
1.（1+y)dx+(x-1)dy=0
2.y’=e^(2x-y),y|x=2 =1
The second one is wrong. It should be 2. Y '= e ^ (2x-y), y | x = 0 = 1

1 DX / 1-x = dy / 1 + y simultaneous integration on both sides can get ln (1 + y) + ln (1-x) = C, that is - XY + X + y + C = 0.2 dy / DX = e ^ 2x / e ^ y, and e ^ YDY = e ^ 2xdx simultaneous integration on both sides 2E ^ y = e ^ 2x + C

differential equation
Differential equations like ay '' (x) + by '(x) + CY (x) = f (x),
We should first find the solution of ay '' (x) + by '(x) + CY (x) = 0, which is the general solution?
Then find a special solution of ay '' (x) + by '(x) + CY (x) = f (x)
Finally =+
Notice that I'm not in the process of asking and solving problems, but asking if I understand the definition of general solution, special solution and solution correctly
If you want to answer, you are welcome to provide English comments, such as general solution and special solution

That's right. The solution of the homogeneous equation is taken as the complementary function, and the special solution of the non-homogeneous equation is added to get the special solution of the non-homogeneous equation
But it's obviously easier to use Laplace transform