Solve y'' + 4y = 8 Step by Step | Undetermined coefficients

Step by step

The question

Question: solve y″ + 4y = 8.

y″ + 4y = 8

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 8.

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 + 4 = 0.

r² + 4 = 0

y″ → r², y′ → r, y → 1

It does not factor nicely, so we use the formula, with a = 1, b = 0, c = 4.

r = −b ± √b² − 4ac2a

Put the numbers in.

r = 0 ± √0² − 4 × 1 × 42 × 1

Under the root: −16.

r = 0 ± √−162

A minus under the root: √(−16) = 4i, because i = √(−1).

√−16 = √16 × √−1 = 4i

r = 0 ± 4i2

Divide both parts by 2: r = ±2i.

Two complex roots.

r = ±2i

Complex roots give cosines and sines.

The real part 0 goes in the e, the 2 goes inside cos and sin.

α ± βi: e^(αx)(C₁cos βx + C₂sin βx)

So the plain part is C₁cos(2x) + C₂sin(2x).

y_c = C₁cos(2x) + C₂sin(2x)

2 Guess the extra part

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

a number → A

So we guess A for the extra part.

A is the numbers we must find.

y_p = A

3 Put the guess in

Its derivative.

y′_p = 0

And once more.

y″_p = 0

Put them into the left side.

0 + 4 × A = 8

Multiply out and put like parts together.

4A = 8

4 Match the parts

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

number parts: 4A = 8

same shape → same number

So A = 2.

A = 2

So the extra part is 2.

y_p = 2

5 Add the parts

Add the parts: y = C₁cos(2x) + C₂sin(2x) + 2.

y = C₁cos(2x) + C₂sin(2x) + 2

6 Check

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

put in y_p: 0 + 4 × 2

It gives 8: the right side.

Right!

= 8

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