Is y = e^(2x) a solution of y' = 3y Step by Step

Step by step

The question

Question: check if y = e^(2x) solves y′ = 3y.

y = e^(2x)

y′ = 3y is a differential equation: an equation with a derivative in it.

y′ = 3y

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 e^(2x) is 2e^(2x).

y′ = 2e^(2x)

The rules we use: the chain rule brings the number in front down.

(e^(kx))′ = ke^(kx)

Careful!

The chain rule!

Times the inside's slope, 2.

Careful: not (e^(2x))′ = e^(2x) ✗

2 Put them in

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

2e^(2x) = 3 × e^(2x)

Multiply out and add like terms.

2e^(2x) = 3e^(2x)

3 Compare the sides

The sides are different.

One number shows it: at x = 1, the left side is 14.78 and the right side is 22.17.

x = 1: left = 14.78, right = 22.17, 14.78 ≠ 22.17

So y = e^(2x) is not a solution.

= no: y = e^(2x) is not a solution

4 Check

Let's check with another number, x = 0.5: 5.44 and 8.15.

Different again.

Right!

x = 0.5: left ≈ 5.44, right ≈ 8.15

Let's look back.

First: find the derivatives.

Then: put them in.

Then: compare the sides.

Last: check.

Answer