Is y = x^4 a solution of xy' = 4y

Step by step

The question

Question: check if y = x⁴ solves xy′ = 4y.

xy′ = 4y is a differential equation: an equation with a derivative in it.

Its answer is not a number.

It is a function y, and it must make the equation true for every x.

1 Find the derivatives

The derivative of x⁴ is 4x³.

The rule we use: the power rule.

2 Put them in

Now put y and y′ into the equation, in place of the letters.

Multiply out and add like terms.

3 Compare the sides

Careful!

A solution must work for every x, not only for one number.

Here both sides are the same expression, so it works for all x.

Both sides are the same, for every x.

So y = x⁴ is a solution.

4 Check

Let's check with one number, x = 1.

A calculator gives 4 on both sides.

Right!

Let's look back.

First: find the derivatives.

Then: put them in.

Then: compare the sides.

Last: check.