Step by step
The question
Question: check if y = x³ solves xy′ = 3y.
y = x3
xy′ = 3y is a differential equation: an equation with a derivative in it.
xy′ = 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 x³ is 3x².
y′ = 3x2
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(3x2) = 3 × x3
Multiply out and add like terms.
3x3 = 3x3
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.
Careful: it must be true for every x, not just one
Both sides are the same, for every x.
So y = x³ is a solution.
= yes: y = x³ is a solution
4 Check
Let's check with one number, x = 1.
A calculator gives 3 on both sides.
Right!
x = 1: left ≈ 3, right ≈ 3
Let's look back.
First: find the derivatives.
Then: put them in.
Then: compare the sides.
Last: check.
Answer
= yes: y = x³ is a solution