Solve dy/dx = 6x^2, y(1) = 5 Step by Step

Step by step

The question

Question: solve dy/dx = 6x², with y(1) = 5.

dydx = 6x²

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

y(1) = 5

1 The idea

dy/dx is only about x.

So y is a function whose derivative is 6x²: an antiderivative.

y′ = 6x²: y is a function whose derivative is 6x²

2 Integrate both sides

Integrate both sides.

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

y = ∫ 6x² 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 = 6 × x³3 + C

Tidy the numbers: y = 2x³ + C.

y = 2x³ + C

Careful!

Don't forget + C.

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

Careful: not y = 2x³ ✗

3 Use the starting value

Put in x = 1 and y = 5.

5 = 2(1)³ + C

Work out the numbers: 2.

5 = 2 + C

To get C alone, take away 2 on both sides.

5 − 2 = C

So C = 3.

C = 3

Put C back.

The answer: y = 2x³ + 3.

y = 2x³ + 3

4 Check

Let's check: the derivative of our answer is 6x².

The right side again.

Right!

y′ = 6x²

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