Solve y' = -4x, y(0) = 7 Step by Step | Integrals as solutions

Step by step

The question

Question: solve y′ = −4x, with y(0) = 7.

y′ = −4x

y(0) = 7 is the starting value: when x is 0, y is 7.

y(0) = 7

1 The idea

y′ is only about x.

So y is a function whose derivative is −4x: an antiderivative.

y′ = −4x: y is a function whose derivative is −4x

2 Integrate both sides

Integrate both sides.

The left side, the integral of y′, is just y.

y = ∫ −4x dx

The power rule backwards: add 1 to the power, divide by the new power.

∫ xⁿ dx = x^{n + 1}n + 1

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

y = −4 × x²2 + C

Tidy the numbers: y = −2x² + C.

y = −2x² + C

Careful!

Don't forget + C.

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

Careful: not y = −2x² ✗

3 Use the starting value

Put in x = 0 and y = 7.

7 = −2(0)² + C

Work out the numbers: 0.

7 = 0 + C

So C = 7.

C = 7

Put C back.

The answer: y = −2x² + 7.

y = −2x² + 7

4 Check

Let's check: the derivative of our answer is −4x.

The right side again.

Right!

y′ = −4x

And at x = 0, y is 7: the starting value.

Right!

y(0) = 7

Let's look back.

First: the idea.

Then: integrate both sides.

Then: use the starting value.

Last: check.

Answer