Solve dy/dx = 3x^2 + 2x, y(2) = 10 Step by Step

Step by step

The question

Question: solve dy/dx = 3x² + 2x, with y(2) = 10.

dydx = 3x² + 2x

y(2) = 10 is the starting value: when x is 2, y is 10.

y(2) = 10

1 The idea

dy/dx is only about x.

So y is a function whose derivative is 3x² + 2x: an antiderivative.

y′ = 3x² + 2x: y is a function whose derivative is 3x² + 2x

2 Integrate both sides

Integrate both sides.

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

y = ∫ (3x² + 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 = 3 × x³3 + 2 × x²2 + C

Tidy the numbers: y = x³ + x² + C.

y = x³ + 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 = x³ + x² ✗

3 Use the starting value

Put in x = 2 and y = 10.

10 = (2)³ + (2)² + C

Work out the numbers: 12.

10 = 12 + C

To get C alone, take away 12 on both sides.

10 − 12 = C

So C = −2.

C = −2

Put C back.

The answer: y = x³ + x² − 2.

y = x³ + x² − 2

4 Check

Let's check: the derivative of our answer is 3x² + 2x.

The right side again.

Right!

y′ = 3x² + 2x

And at x = 2, y is 10: the starting value.

Right!

y(2) = 10

Let's look back.

First: the idea.

Then: integrate both sides.

Then: use the starting value.

Last: check.

Answer