Solve dy/dx = -3e^(-x), y(0) = 1 Step by Step

Step by step

The question

Question: solve dy/dx = −3e^(−x), with y(0) = 1.

dydx = −3e^(−x)

y(0) = 1 is the starting value: when x is 0, y is 1.

y(0) = 1

1 The idea

dy/dx is only about x.

So y is a function whose derivative is −3e^(−x): an antiderivative.

y′ = −3e^(−x): y is a function whose derivative is −3e^(−x)

2 Integrate both sides

Integrate both sides.

The left side, the integral of dy/dx, is just y.

y = ∫ −3e^(−x) dx

The rule: divide by the number in front of x.

∫ e^kx dx = e^kxk

Each term by its rule, and one + C: the number we cannot see after a derivative.

y = −3 × e^(−x)(−1) + C

Tidy the numbers: y = 3e^(−x) + C.

y = 3e^(−x) + C

Careful!

Don't forget + C.

Every number C gives another solution, and we need C to use the starting value.

Careful: not y = 3e^(−x) ✗

3 Use the starting value

Put in x = 0 and y = 1.

1 = 3e^(0) + C

e^(0) = 1

Work out the numbers: 3.

1 = 3 + C

To get C alone, take away 3 on both sides.

1 − 3 = C

So C = −2.

C = −2

Put C back.

The answer: y = 3e^(−x) − 2.

y = 3e^(−x) − 2

4 Check

Let's check: the derivative of our answer is −3e^(−x).

The right side again.

Right!

y′ = −3e^(−x)

And at x = 0, y is 1: the starting value.

Right!

y(0) = 1

Let's look back.

First: the idea.

Then: integrate both sides.

Then: use the starting value.

Last: check.

Answer