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Unit 1 — First Order Equations

Separable DEs

Build a model first, then learn the method that matches it.

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Big idea

A separable equation is one where you can pull the variables apart.
You rearrange the equation so the y-stuff is with dydy and the t-stuff is with dtdt, then integrate both sides.

IDEA

What makes an equation separable?

The pattern to look for

You want to rewrite the DE so each variable lives on its own side.

A common separable form is:

dydt=g(t)h(y)\frac{dy}{dt} = g(t)h(y)

Then you separate:

dyh(y)=g(t) dt\frac{dy}{h(y)} = g(t)\,dt

Quick check

choose one
Which equation is already in a separable-friendly form?

WALKTHROUGH

See the separation happen

Interactive: separate step by step

Watch how a separable equation gets rewritten into something you can integrate.

Start

Begin with dydt=ty\frac{dy}{dt} = ty.

METHOD

The separable equation workflow

1) Separate variables

Move all y terms with dydy and all t terms with dtdt.

2) Integrate both sides

Once separated, integrate each side like a regular calculus problem.

3) Simplify / solve for y

After integrating, rearrange if needed to get an explicit formula for y.

4) Use the initial condition

If the problem gives something like y(0)=2y(0)=2, plug it in to find the constant.

PRACTICE

Practice: do the method

Practice progress

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Quick check

choose one
If dydt=ty\frac{dy}{dt} = ty, what is the correct separated form?

Quick check

choose one
After separating dyy=t dt\frac{dy}{y} = t\,dt, what do you get after integrating?

3) Initial condition

keyword check

Why do we use an initial condition after integrating a separable equation?

Matched concepts: 0/2

4) Explain the logic

keyword check

In your own words, what is the main idea behind the separable equations method?

Matched concepts: 0/2

WHAT'S NEXT

Separation is a solving method

Unlike slope fields, separable equations give you a direct way to compute formulas for solutions. Later, you’ll connect this method to growth, decay, and logistic models.

← Back to roadmapNext: Growth & Decay →