Step by step
The question
Question: check if y = x² solves y′ = 2y.
y = x2
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 x² is 2x.
y′ = 2x
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.
2x = 2 × x2
Multiply out and add like terms.
2x = 2x2
3 Compare the sides
The sides are different.
One number shows it: at x = 2, the left side is 4 and the right side is 8.
x = 2: left = 4, right = 8, 4 ≠ 8
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 is not a solution.
= no: y = x² is not a solution
4 Check
Let's check with another number, x = 0.5: 1 and 0.5.
Different again.
Right!
x = 0.5: left ≈ 1, right ≈ 0.5
Let's look back.
First: find the derivatives.
Then: put them in.
Then: compare the sides.
Last: check.
Answer
= no: y = x² is not a solution