Solve y'' - 3y' = 6 Step by Step | Undetermined coefficients

Step by step

The question

Question: solve y″ − 3y′ = 6.

y″ − 3y′ = 6

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 6.

y = y_c + y_p: the plain part + one extra solution

1 Solve with 0 on the right

Try y = e^(rx): the characteristic equation is r² − 3r = 0.

r² − 3r = 0

y″ → r², y′ → r, y → 1

Every term has an r: take it out.

r(r − 3) = 0

A product is 0 when one part is 0: r = 0 or r = 3.

r = 0 or r = 3

Each root gives a solution e^(rx).

Any mix of the two works, so we add them with two numbers, C₁ and C₂.

two roots r₁, r₂: C₁e^(r₁x) + C₂e^(r₂x)

So the plain part is C₁ + C₂e^(3x).

y_c = C₁ + C₂e^(3x)

2 Guess the extra part

The extra part has the same shape as the right side: just a number.

a number → A

But 1 is already in the plain part: it gives 0 on the left.

So we multiply it by x.

already in y_c → times x

Careful!

That one is already in the plain part, so it can only give 0.

Times x.

Careful: not y_p = A ✗

So we guess Ax for the extra part.

A is the numbers we must find.

y_p = Ax

3 Put the guess in

Its derivative.

y′_p = A

And once more.

y″_p = 0

Put them into the left side.

0 − 3 × A = 6

Multiply out and put like parts together.

−3A = 6

4 Match the parts

The two sides must match, part by part: the same amount of each shape.

number parts: −3A = 6

same shape → same number

So A = −2.

A = −2

So the extra part is −2x.

y_p = −2x

5 Add the parts

Add the parts: y = C₁ + C₂e^(3x) − 2x.

y = C₁ + C₂e^(3x) − 2x

6 Check

Let's check the extra part: put −2x into the left side.

put in y_p: 0 − 3(−2)

It gives 6: the right side.

Right!

= 6

Let's look back.

First: solve with 0 on the right.

Then: guess the extra part.

Then: put the guess in.

Then: match the parts.

Then: add the parts.

Last: check.

Answer