Substitution for dy/dx = g(ax + by + c) (When RHS depends only on linear combo ax+by+c, set z = ax+by+c. Then)

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When RHS depends only on linear combo ax+by+c, set z = ax+by+c. Then dz/dx = a + b dy/dx, substitute dy/dx from DE to get separable or simpler DE in z and x.

What each symbol means

SymbolWhat it stands for
zLinear combination ax + by + c.

When to use this

b ≠ 0 typically needed to express dy/dx from dz/dx cleanly; b=0 is already function of x alone.