Step by step
The question
Question: check if y = e^(−2x) solves y′ = −2y.
y = e^(−2x)
y′ = −2y is a differential equation: an equation with a derivative in it.
y′ = −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 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) = −2 × e^(−2x)
Multiply out and add like terms.
−2e^(−2x) = −2e^(−2x)
3 Compare the sides
Both sides are the same, for every x.
So y = e^(−2x) is a solution.
= yes: y = e^(−2x) is a solution
4 Check
Let's check with one number, x = 1.
A calculator gives −0.27 on both sides.
Right!
x = 1: left ≈ −0.27, right ≈ −0.27
Let's look back.
First: find the derivatives.
Then: put them in.
Then: compare the sides.
Last: check.
Answer
= yes: y = e^(−2x) is a solution