I spent two weeks staring at Bernoulli differential equations thinking I was just bad at math. Turns out I was missing one mental shift.

Most textbooks throw the formula at you — y' + P(x)y = Q(x)yⁿ — and then jump into substitution without explaining why it works.

Here's the thing: the substitution v = y¹⁻ⁿ isn't random. It's specifically designed to kill the nonlinearity. Once you see that every Bernoulli equation is just a linear ODE in disguise, the entire process becomes mechanical:

1. Divide everything by yⁿ

2. Substitute v = y¹⁻ⁿ

3. Now you have a standard first-order linear ODE

4. Solve with integrating factor like you already know how to

That's it. The "hard" part was never the solving — it was not understanding what the substitution was doing.

I've been documenting shortcuts like this for every intimidating topic in upper-level math. Differential equations, linear algebra, real analysis — all of it broken down the way I wish my professors explained it.

If you're a STEM student drowning in coursework, The Student Logic Vault is where I put all of this. Handwritten, step-by-step guides that actually make sense.