Is y = x^2 + 2x a solution of xy' = 2y Step by Step

Step by step

The question

Question: check if y = x² + 2x solves xy′ = 2y.

y = x² + 2x

xy′ = 2y is a differential equation: an equation with a derivative in it.

xy′ = 2y

Its answer is not a number.

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

differential equation: an equation with a derivative

its answer: a function y, true for every x

1 Find the derivatives

The derivative of x² + 2x is 2x + 2.

y′ = 2x + 2

The rule we use: the power rule.

(xⁿ)′ = nxⁿ⁻¹

2 Put them in

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

x(2x + 2) = 2(x² + 2x)

Multiply out and add like terms.

2x² + 2x = 2x² + 4x

3 Compare the sides

The sides are different.

One number shows it: at x = 1, the left side is 4 and the right side is 6.

x = 1: left = 4, right = 6, 4 ≠ 6

Careful!

A solution must work for every x.

One x where it fails is enough to say no.

Careful: it must be true for every x

So y = x² + 2x is not a solution.

= no: y = x² + 2x is not a solution

4 Check

Let's check with another number, x = 0.5: 1.5 and 2.5.

Different again.

Right!

x = 0.5: left ≈ 1.5, right ≈ 2.5

Let's look back.

First: find the derivatives.

Then: put them in.

Then: compare the sides.

Last: check.

Answer