Separation of Variables (College Board AP® Calculus AB)

Revision Note

Author

Roger B

Expertise

Maths

Separation of variables

What is separation of variables?

Separation of variables can be used to solve certain types of first order differential equations
Look out for equations of the form $\frac{d y}{d x} = g (x) h (y)$
- I.e. $\frac{d y}{d x}$ is equal to a function of $x$ multiplied by a function of $y$
- Be careful – the ‘function of $x$ ’ $g (x)$ may just be a constant!
  - For example in $\frac{d y}{d x} = 6 y$ , $g (x) = 6$ and $h (y) = y$
If the equation is in that form
- then you can use separation of variables to try to solve it

How do I solve a differential equation using separation of variables?

STEP 1
Rearrange the equation into the form $(\frac{1}{h (y)}) \frac{d y}{d x} = g (x)$
- E.g. $\frac{d y}{d x} = - x y^{3} \Rightarrow - \frac{1}{y^{3}} \frac{d y}{d x} = x$
  - $g (x) = x, h (y) = - y^{3}$
STEP 2
Integrate both sides with respect to $x$
- This changes the equation into the form $\int \frac{1}{h (y)} d y = \int g (x) d x$
- E.g. $- \frac{1}{y^{3}} \frac{d y}{d x} = x \Rightarrow \int \frac{- 1}{y^{3}} d y = \int x d x$
  - You can think of this step as ‘multiplying the $d x$ across and integrating both sides’
    - Mathematically that’s not quite what is happening, but it will get you the right answer here!
STEP 3
Work out the integrals on both sides of the equation
- Don’t forget to include a constant of integration
  - You only need one constant of integration, even though there are two integrals
- E.g. $\int \frac{- 1}{y^{3}} d y = \int x d x \Rightarrow - \int y^{- 3} d y = \int x d x \Rightarrow \frac{1}{2} y^{- 2} = \frac{1}{2} x^{2} + C$
STEP 4
Rearrange the solution
- E.g. $\frac{1}{2} y^{- 2} = \frac{1}{2} x^{2} + C \Rightarrow \frac{1}{y^{2}} = x^{2} + 2 C \Rightarrow y^{2} = \frac{1}{x^{2} + 2 C}$
- Note that you won't always be able to rewrite the solution in $y = f (x)$ form
  - In this case $y = \sqrt{\frac{1}{x^{2} + 2 C}}$ is not correct, because the solutions also include the $y = - \sqrt{\frac{1}{x^{2} + 2 C}}$ option
  - If an exam question requires the answer in a particular form, be sure to rearrange into that form
- Also note that $2 C$ is just another arbitrary integration constant
  - So $y^{2} = \frac{1}{x^{2} + C}$ would be a 'neater' way to write the solution
This method gives the general solution to the differential equation
- For finding the particular solution, see the 'Particular Solutions' study guide

Worked Example

Use separation of variables to solve the differential equation $\frac{d y}{d x} = \frac{e^{x} + 4 x}{3 y^{2}}$ .

Answer:

Separate the variables, getting all the $y$ terms on the $\frac{d y}{d x}$ side and all the $x$ terms on the other side

$3 y^{2} \frac{d y}{d x} = e^{x} + 4 x$

Integrate both sides with respect to $x$

$\int 3 y^{2} d y = \int (e^{x} + 4 x) d x$

Integrate (and don't forget a constant of integration!)

$y^{3} = e^{x} + 2 x^{2} + C$

This can be written in $y = f (x)$ form by taking the cube root of both sides

$y = \sqrt[3]{e^{x} + 2 x^{2} + C}$

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