Solve y' = 4x^3 - 2x, y(1) = 5 Step by Step

Step by step

The question

Question: solve y′ = 4x³ − 2x, with y(1) = 5.

y′ = 4x³ − 2x

y(1) = 5 is the starting value: when x is 1, y is 5.

y(1) = 5

1 The idea

y′ is only about x.

So y is a function whose derivative is 4x³ − 2x: an antiderivative.

y′ = 4x³ − 2x: y is a function whose derivative is 4x³ − 2x

2 Integrate both sides

Integrate both sides.

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

y = ∫ (4x³ − 2x) 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 × x⁴4 − 2 × x²2 + C

Tidy the numbers: y = x⁴ − x² + C.

y = x⁴ − x² + C

Careful!

Don't forget + C.

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

Careful: not y = x⁴ − x² ✗

3 Use the starting value

Put in x = 1 and y = 5.

5 = (1)⁴ − (1)² + C

Work out the numbers: 0.

5 = 0 + C

So C = 5.

C = 5

Put C back.

The answer: y = x⁴ − x² + 5.

y = x⁴ − x² + 5

4 Check

Let's check: the derivative of our answer is 4x³ − 2x.

The right side again.

Right!

y′ = 4x³ − 2x

And at x = 1, y is 5: the starting value.

Right!

y(1) = 5

Let's look back.

First: the idea.

Then: integrate both sides.

Then: use the starting value.

Last: check.

Answer