Topic 7.6 Notes – Finding General Solutions Using Separation of Variables

What a General Solution to a Differential Equation Is

A differential equation is an equation that involves a function and its derivative, usually written as
$$\frac{dy}{dx} = \text{(something)}$$

A solution is a function $y = f(x)$ that makes the equation true when you plug it in.

A general solution:

• Contains an arbitrary constant $C$
• Represents a whole family of functions
• Comes from antidifferentiation, so it must include “+ C”
• Does not use an initial condition (that’s the next topic)

For example, if
$$\frac{dy}{dx} = 4x,$$ then integrating gives
$$y = 2x^2 + C.$$

Every different value of $C$ gives a different curve.

Here’s what that family looks like visually:

Family of solutions $y = 2x^2 + C$

All of those curves have derivative $4x$. That’s what makes them all solutions.

Separable Differential Equations

Some differential equations can be rewritten so that all the $y$-stuff is on one side and all the $x$-stuff is on the other.

A first-order equation is separable if it can be written as
$$\frac{dy}{dx} = g(x)h(y).$$

That product structure is the clue.

If you can rearrange it into
$$\text{(function of } y)\, dy = \text{(function of } x)\, dx,$$ then it’s separable.

Example that is separable:
$$\frac{dy}{dx} = 3x y^2$$

You can rewrite it as:
$$\frac{1}{y^2} dy = 3x \, dx.$$

Example that is not separable:
$$\frac{dy}{dx} = y + x.$$

There’s no way to split that into pure $x$ times pure $y$. On a quiz, recognizing when you can’t separate is just as important.

The Separation of Variables Process

Once it’s separable, the steps are mechanical but you need to be clean.

1. Rewrite so it looks like $\frac{dy}{dx} = g(x)h(y)$.
2. Separate variables
   Move all $y$’s with $dy$, all $x$’s with $dx$.
3. Integrate both sides
   $$\int (\text{y-expression})\, dy = \int (\text{x-expression})\, dx$$
4. Add one constant $C$
   (You only need one. Combine constants.)
5. Solve for $y$ if possible.

Let’s walk one through.

Solve:
$$\frac{dy}{dx} = x y$$

Separate:
$$\frac{1}{y} dy = x\, dx$$

Integrate:
$$\int \frac{1}{y} dy = \int x\, dx$$

$$\ln|y| = \frac{1}{2}x^2 + C$$

Exponentiate:

$$|y| = e^{\frac{1}{2}x^2 + C}$$

Rewrite $e^C$ as a new constant:

$$y = C e^{\frac{1}{2}x^2}$$

That’s the general solution.

Integration Patterns That Show Up a Lot

Power rule forms

If you see $y^n$ or $x^n$, use the power rule.

Be careful when dividing by $y$.
$\frac{1}{y} = y^{-1}$, which gives a logarithm.

The $\frac{1}{y}$ situation

$$\int \frac{1}{y} dy = \ln|y|$$

Two things students lose points on:

• Forgetting the absolute value
• Forgetting to exponentiate to solve for $y$

If you get:
$$\ln|y| = \text{something},$$ you almost always end with:
$$y = C e^{(\text{something})}.$$

This structure shows up constantly on FRQs.

What This Looks Like on Tests

• On free response, you must show the separation and both integrals clearly.
• On multiple choice, they may give you four possible families and ask which satisfies the differential equation.
• Sometimes they test recognition by giving something that looks separable but isn’t.

Also, if you stop at an implicit answer when it’s easy to solve for $y$, you may lose a point. Always check whether you can cleanly isolate $y$.