Topic 7.6 Notes – Finding General Solutions Using Separation of Variables

What a General Solution to a Differential Equation Is

A differential equation connects a function $y$ and its derivative $\frac{dy}{dx}$.

A solution is any differentiable function that makes the equation true when you plug it in.

A general solution is a family of solutions. It includes an arbitrary constant $C$.

Why is there a constant?

• When you differentiate, constants disappear.
• When you antidifferentiate, you must add them back.

Example idea:

If $$\frac{dy}{dx} = 4x$$

then integrating gives $$y = 2x^2 + C$$

That $C$ represents infinitely many possible curves, all with slope $4x$.

On quizzes and the AP exam, “find the general solution” means:

• Separate (if possible)
• Integrate
• Include $+C$

Initial conditions come later. Right now, you’re just finding the whole family.

What Makes a Differential Equation Separable

A first-order differential equation is separable if it can be written as

$$\frac{dy}{dx} = g(x)h(y)$$

That means:

• The right side is a product of something involving only $x$
• And something involving only $y$

If you can rewrite it so all the $y$’s go with $dy$ and all the $x$’s go with $dx$, you’re good.

Examples That Are Separable

• $\frac{dy}{dx} = x e^y$
• $\frac{dy}{dx} = \frac{3x^2}{y}$
• $\frac{dy}{dx} = y(5 - x)$

Each can be rearranged into a pure $x$-side and pure $y$-side.

Example That Is Not Separable

• $\frac{dy}{dx} + xy = y^2$

You cannot rewrite that as a clean product $g(x)h(y)$. If you can’t isolate into “all x’s times all y’s,” separation won’t work.

A quick mental check: can I physically move every $y$ next to $dy$ and every $x$ next to $dx$? If yes, it’s separable.

How to Solve by Separation of Variables

Let’s walk through a full example:

$$\frac{dy}{dx} = x^3 y$$

Step 1. Separate

Divide both sides by $y$:

$$\frac{1}{y} dy = x^3 dx$$

Now the variables are separated.

Step 2. Integrate Both Sides

$$\int \frac{1}{y} dy = \int x^3 dx$$

Left side: $$\ln |y|$$

Right side: $$\frac{x^4}{4}$$

So:

$$\ln |y| = \frac{x^4}{4} + C$$

Step 3. Solve for $y$

Exponentiate both sides:

$$|y| = e^{\frac{x^4}{4} + C}$$

Use exponent rules:

$$|y| = e^C e^{\frac{x^4}{4}}$$

Since $e^C$ is just another constant, rewrite:

$$y = C e^{\frac{x^4}{4}}$$

That’s the general solution.

What General Solutions Usually Look Like

After separation and integration, solutions tend to fall into patterns:

• Logarithmic form
  Comes from $\int \frac{1}{y} dy$
  You’ll see $\ln |y| = \text{something}$
• Exponential form
  Happens after exponentiating
  $y = Ce^{g(x)}$
• Power form
  From integrals like $\int y \, dy$
  You might get something like
  $y^2 = x^3 + C$

If the derivative is proportional to $y$, expect an exponential answer.

Common Mistakes and Exam Traps

These cost points every year:

• Forgetting to move all y’s with $dy$
• Leaving two constants instead of combining into one $C$
• Dropping the absolute value in $\ln |y|$
• Forgetting to include $+C$
• Not solving for $y$ when it’s reasonable to do so

Also, if both sides produce constants after integration, combine them immediately. The grader expects a single arbitrary constant.

On free response, clean algebra matters. If you exponentiate, simplify $e^C$ into $C$. Don’t leave it messy.