Solve y'' + 4y = 3cos(x) Step by Step | Undetermined coefficients

Step by step

The question

Question: solve y″ + 4y = 3cos x.

y″ + 4y = 3cos x

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 3cos x.

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: a cosine and a sine, both, with the same 1x inside.

sin or cos → A cos(bx) + B sin(bx)

Careful!

Both cos and sin are needed: each derivative turns one into the other.

Careful: not y_p = Acos x ✗

So we guess Acos x + Bsin x for the extra part.

A and B are the numbers we must find.

y_p = Acos x + Bsin x

3 Put the guess in

Its derivative.

y′_p = −Asin x + Bcos x

And once more.

y″_p = −Acos x − Bsin x

Put them into the left side.

−Acos x − Bsin x + 4(Acos x + Bsin x) = 3cos x

Multiply out and put like parts together.

3Acos x + 3Bsin x = 3cos x

4 Match the parts

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

cos x parts: 3A = 3

sin x parts: 3B = 0

same shape → same number

So A = 1.

A = 1

So B = 0.

B = 0

So the extra part is cos x.

y_p = cos x

5 Add the parts

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

y = C₁cos(2x) + C₂sin(2x) + cos x

6 Check

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

put in y_p: −cos x + 4 × cos x

It gives 3cos x: the right side.

Right!

= 3cos x

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