求助：证明替换z=ax+by+c可将y′=f(ax+by+c)化为可分离变量方程

Hey there! Let's break down this proof step by step—this substitution trick is a go-to for simplifying these kinds of ODEs, so it's worth walking through clearly.

First, a quick refresher: a separable differential equation is one that can be rearranged so all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the other. For our case, we want to get it into a form where we can separate ( z ) and ( x ).

We start with the given ODE:

( y' = f(ax + by + c) )

Step 1: Compute the derivative of the substitution #

Let’s define our substitution ( z = ax + by + c ). To link this to ( y' ), we’ll take the derivative of ( z ) with respect to ( x ):

dz/dx = d/dx(ax + by + c) = a + b \cdot y'

Now solve this equation for ( y' ) (note: we’re assuming ( b \neq 0 ) here—we’ll cover the ( b=0 ) case later):

y' = (dz/dx - a)/b

Step 2: Substitute into the original ODE #

Replace ( y' ) in the original equation with the expression we just found. Also, notice that ( ax + by + c = z ), so ( f(ax + by + c) = f(z) ):

(dz/dx - a)/b = f(z)

Step 3: Rearrange to get a separable form #

Multiply both sides by ( b ) to eliminate the denominator:

dz/dx - a = b \cdot f(z)

Add ( a ) to both sides to isolate the derivative term:

dz/dx = b \cdot f(z) + a

Let’s define a new function ( g(z) = b \cdot f(z) + a ) (since ( a ) and ( b ) are constants, this is just a function of ( z )). Now our equation becomes:

dz/dx = g(z)

This is clearly separable! We can rearrange it to:

1/g(z) dz = dx

(Note: If ( g(z) = 0 ), those correspond to constant solutions ( z = k ) where ( b \cdot f(k) + a = 0 ). Translating back to ( y ), that gives ( y = (k - ax - c)/b ), which are constant solutions to the original ODE.)

Step 4: Handle the edge case when ( b = 0 ) #

If ( b = 0 ), the original equation simplifies to ( y' = f(ax + c) )—this is already a separable equation! We can directly write:

dy = f(ax + c) dx

No substitution is even needed here, but the trick still holds (since substituting ( z = ax + c ) would just confirm separability).

Wrapping up #

By using the substitution ( z = ax + by + c ), we’ve transformed the original "combined variable" ODE into a straightforward separable equation. This is a classic technique that makes solving these types of problems much more manageable—once you make the substitution, it’s just algebraic rearrangement to get things into a form you can integrate.

内容的提问来源于stack exchange，提问作者Agboola Olumuyiwa Damilare