There are two kinds of differential equations in the reducible higher order differential equations: y '' = f (x, y ') and y' '= f (y, y') The former equation can be solved by Y '= P, then y' = DP / DX = P ', while the latter equation can be solved by Y' = P, then y '= P * DP / dy. for example: YY' - y '^ 2 = 0, this equation is expressed by Y' = P, y '= P * DP / dy. why is y' = 1 + y '^ 2 solved by Y' = P, y '= P'? I know that, but the examples I gave are all short of X-type. Why are they different? | Math enthusiast | Mathematical art

There are two kinds of differential equations in the reducible higher order differential equations: y '' = f (x, y ') and y' '= f (y, y') The former equation can be solved by Y '= P, then y' = DP / DX = P ', while the latter equation can be solved by Y' = P, then y '= P * DP / dy. for example: YY' - y '^ 2 = 0, this equation is expressed by Y' = P, y '= P * DP / dy. why is y' = 1 + y '^ 2 solved by Y' = P, y '= P'? I know that, but the examples I gave are all short of X-type. Why are they different?

In brief:
The former P is a function of X and the latter P is a function of Y
The former P & # 39; is the derivation of X, and the latter P & # 39; is the derivation of Y
See the picture below

RELATED INFORMATIONS

1. The reducible higher order differential equation XY '+ 1 = y' ^ 2
2. The second derivative of y to x equals ax & sup3; + B I know that first let the first derivative of y be equal to u, and then change it into the first derivative of u by substitution, but then a polynomial with the square of u equal to x appeared, and I can't do it any more Seek expert solution Wrong, it's equal to ay & sup3; + B Otherwise it would be too simple~
3. On the problem of solving second order non homogeneous linear differential equation with constant coefficient in postgraduate entrance examination mathematics It is known that the equation is a second-order non-homogeneous linear differential equation with constant coefficients, and it has two special solutions: Y1 = cos2x-1 / 4xsin2x, y2 = sin2x-1 / 4xsin2x. Now the expression of the equation is required. Let the general solution of the equation be y = C1 * cos2x + C2 * sin2x - 1 / 4sin2x, where C1 and C2 are arbitrary constants, Cos2x and sin2x should be a special solution of the corresponding homogeneous differential equation, - 1 / 4sin2x should be a special solution of this equation, but the problem does not give these two conditions, ah, hope to have a great God to help solve, thank you!
4. A mathematical differential equation problem (x + 1) y '+ 1 = 2 (to the - y power of E) Ask for his general understanding
5. Does the content of the third postgraduate entrance examination of mathematics include the higher order differential equation that can reduce the price
6. For the point (x0, Y0, Z0), t tends to 0, and the function f () satisfies f（x0+t,y,z)=f(x0,y0,z0)*P(y-y0,z-z0）； Where p () is a two-dimensional normal distribution function related to y-y0 and z-z0, Given the initial value of F (x0, Y0, Z0), I want to find the value of any f (x1, y, z) at x = x1, as long as I think about it If you can do it, I'll give you my share The fairy on the first floor f(x0,y0,z0)*P(y-y0,z-z0）-f（x0,y,z) =F (x0, Y0, Z0) (P (y-y0, z-z0) - 1) is not very right, I'm changing f (x0 + T, y, z) = f (x0, Y0, Z0) * P (y-y0, z-z0) × exp (- at); a is a known constant
7. Solving differential equation y '= cosx / Y
8. Which differential equation is y = cosx
9. Solve the differential equation (x ^ 2-1) y '+ 2XY cosx = 0, thank you very much! (x ^ 2-1) y & # 39; + 2XY cosx = 0, the garbled part is the derivative of Y, detailed process
10. Differential equation solution: y '- ysinx - y ^ 2 + cosx = 0
11. Find the special solution of the differential equation 4Y "- 12Y '+ 9y = 0 satisfying the initial conditions y ▏ y = 0 = 1, y' ▏ x = 0 = 1 The comma above is an apostrophe
12. Y "+ 9y = xsin3x to solve differential equations
13. Solving differential equation y "+ 9y = sin3x Thank you very much. Let's go back quickly. The answer is y = c1sin3x + c2sin3x-1 / 6xsin3x
14. Find the solution of the differential equation y ^ (4) + y '' '+ y' + y = 0,
15. The general solution of the differential equation y "+ 8y '+ 16y = 0 is?
16. Find the general solution of the differential equation y '' '+ 8y = 0
17. Solving differential equation y * y '' - (y ') ^ 2-y ^ 2 * y' = 0
18. Special solution of differential equation y '= y satisfying initial condition y (0) = 1 Right away
19. Find the solution of the differential equation y '' = y'e ^ y satisfying the condition y (0) = 0, y '(0) = 1
20. Special solution of differential equation y '+ y = 1 satisfying initial condition y (0) = 0