Solve 2y' + 4y = 8, y(0) = 3 Step by Step

Step by step

The question

Question: solve 2y′ + 4y = 8, with y(0) = 3.

2y′ + 4y = 8

y(0) = 3 is the starting value.

y(0) = 3

1 Standard form

Linear means this form: y′ + P y = Q.

P and Q have only x.

linear: y′ + P(x) y = Q(x)

Divide everything by 2, so y′ is alone at the front.

y′ + 2y = 4

÷ 2

Here P = 2 and Q = 4.

P = 2, Q = 4

2 The integrating factor

The integrating factor is e to the integral of P.

Multiplying by it makes the left side one derivative.

integrating factor: μ = e^(∫P dx)

First the integral of P: the integral of 2 is 2x.

∫ 2 dx = 2x

So the integrating factor is e^(2x).

No + C needed here: any one factor works.

μ = e^(2x)

3 Multiply both sides

Multiply both sides by e^(2x).

Every term, on both sides.

e^(2x)y′ + 2e^(2x)y = 4e^(2x)

Careful!

The right side is multiplied by e^(2x) too.

Both sides, always.

Careful: not (e^(2x)y)′ = 4 ✗

4 One derivative on the left

By the product rule, the derivative of e^(2x) × y is exactly our left side.

(μy)′ = μy′ + μ′y

μ′ = 2e^(2x)

So the left side is just (e^(2x)y)′: one derivative.

(e^(2x)y)′ = 4e^(2x)

5 Integrate both sides

Integrate both sides.

The integral of a derivative is the function itself.

e^(2x)y = ∫ (4e^(2x)) dx

On the right: 2e^(2x), and + C.

e^(2x)y = 2e^(2x) + C

6 Solve for y

Divide both sides by e^(2x).

y = 2e^(2x) + Ce^(2x)

Divide each term.

Dividing by e to a power is the same as e to minus that power.

y = 2 + Ce^(−2x)

C over e^(2x) = Ce^(−2x)

7 Use the starting value

Put in x = 0 and y = 3.

e⁰ = 1.

3 = 2 + C × 1

So C = 1.

C = 1

So y = 2 + e^(−2x).

y = 2 + e^(−2x)

8 Check

Let's check: put y back in.

First y′ is −2e^(−2x).

y′ = −2e^(−2x)

Then y′ + 2y.

y′ + 2y = −2e^(−2x) + 2(2 + e^(−2x))

The e parts cancel, and we get 4: Q again.

Right!

= 4

Let's look back.

First: standard form.

Then: the integrating factor.

Then: multiply both sides.

Then: one derivative on the left.

Then: integrate both sides.

Then: solve for y.

Then: use the starting value.

Last: check.

Answer