Step by step
The question
Question: solve 3y″ − 4y′ + y = 0.
1 Try y = e to the rx
The derivatives of e to the rx are e to the rx again, times r.
So we try y = e^(rx), and look for the right r.
Its derivative: r e^(rx).
Once more: r² e^(rx).
2 The characteristic equation
Put them into the equation.
Every term has e^(rx): take it out.
e to a power is never 0.
So the bracket must be 0.
3r² − 4r + 1 = 0: the characteristic equation.
y″ became r², y′ became r, y became 1.
3 Solve for r
It does not factor nicely, so we use the formula, with a = 3, b = −4, c = 1.
Put the numbers in.
Under the root: 4.
Minus gives 1/3, plus gives 1.
Careful!
The whole top is over 2a, not only the root.
4 Write the solution
Each root gives a solution e^(rx).
Any mix of the two works, so we add them with two numbers, C₁ and C₂.
So y = C₁e^(x/3) + C₂eˣ.
5 Check
Let's check: each root must make 3r² − 4r + 1 = 0.
r = 1/3 gives 0.
And r = 1 gives 0.
Right!
Let's look back.
First: try y = e to the rx.
Then: the characteristic equation.
Then: solve for r.
Then: write the solution.
Last: check.