Is y = 2e^(-x) + 3 a solution of y' + y = 3 Step by Step

Step by step

The question

Question: check if y = 2e^(−x) + 3 solves y′ + y = 3.

y = 2e^(−x) + 3

y′ + y = 3 is a differential equation: an equation with a derivative in it.

y′ + y = 3

Its answer is not a number.

It is a function y, and it must make the equation true for every x.

differential equation: an equation with a derivative

its answer: a function y, true for every x

1 Find the derivatives

The derivative of 2e^(−x) + 3 is −2e^(−x).

y′ = −2e^(−x)

The rules we use: the chain rule brings the number in front down.

(e^(kx))′ = ke^(kx)

2 Put them in

Now put y and y′ into the equation, in place of the letters.

−2e^(−x) + (2e^(−x) + 3) = 3

Multiply out and add like terms.

3 = 3

3 Compare the sides

Careful!

A solution must work for every x, not only for one number.

Here both sides are the same expression, so it works for all x.

Careful: it must be true for every x, not just one

Both sides are the same, for every x.

So y = 2e^(−x) + 3 is a solution.

= yes: y = 2e^(−x) + 3 is a solution

4 Check

Let's check with one number, x = 1.

A calculator gives 3 on both sides.

Right!

x = 1: left ≈ −0.74 + 3.74 = 3, right ≈ 3

Let's look back.

First: find the derivatives.

Then: put them in.

Then: compare the sides.

Last: check.

Answer