Step by step
The question
Question: check if y = 3e^(2x) − e^(3x) solves y″ − 5y′ + 6y = 0.
y″ − 5y′ + 6y = 0 is a differential equation: an equation with a derivative in it.
Its answer is not a number.
It is a function y, and it must make the equation true for every x.
1 Find the derivatives
The derivative of 3e^(2x) − e^(3x) is 6e^(2x) − 3e^(3x).
The rules we use: the chain rule brings the number in front down.
Careful!
The chain rule!
Times the inside's slope, 2.
Once more: y″ is the derivative of y′.
It is 12e^(2x) − 9e^(3x).
2 Put them in
Now put y, y′ and y″ into the equation, in place of the letters.
Multiply out and add like terms.
On the left, everything cancels: 0.
3 Compare the sides
Both sides are the same, for every x.
So y = 3e^(2x) − e^(3x) is a solution.
4 Check
Let's check with one number, x = 1.
A calculator gives 0 on both sides.
Right!
Let's look back.
First: find the derivatives.
Then: put them in.
Then: compare the sides.
Last: check.