Solve dy/dx = 2xy, y(0) = 3 Step by Step

Step by step

The question

Question: solve dy/dx = 2xy, with y(0) = 3.

dydx = 2xy

y(0) = 3 is the starting value: when x is 0, y is 3.

y(0) = 3

The right side splits into an x part, 2x, times a y part, y.

That is a separable equation.

separable: 2x × y

1 Separate x and y

Divide both sides by y, so y is only on the left.

1y × dydx = 2x

÷ y both sides

Now multiply both sides by dx.

All the y parts are with dy, all the x parts with dx.

1y dy = 2x dx

2 Integrate both sides

Put an integral sign on both sides.

∫ 1y dy = ∫ 2x dx

On the left: ln |y|.

On the right: x².

One + C is enough: the two numbers join into one.

ln |y| = x² + C

The integral of 1/y is the natural log of the size of y.

∫ 1y dy = ln |y|

3 Solve for y

To undo ln, use e on both sides: e to the ln of the size of y is just |y|.

|y| = e^(x² + C)

A plus in the power splits into a times: e^(x² + C) = e^C × e^(x²).

|y| = e^(C) × e^(x²)

e^(a + b) = e^(a) × e^(b)

Careful!

The C is not added at the end.

e^C is a number that multiplies.

Careful: not y = e^(x²) + C ✗

e^C is just some number.

With the ± from |y|, call it A: any number.

y = Ae^(x²)

A = ±e^(C) (any number)

4 Use the starting value

Put in x = 0 and y = 3.

The power is 0 there, and e⁰ = 1.

3 = Ae^(0)

So A = 3.

A = 3

So y = 3e^(x²).

y = 3e^(x²)

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ≈ 8.155.

x = 1: y ≈ 8.155

The slope between two very close points: about 16.31.

slope: y(1.001) − y(0.999)0.002 ≈ 16.31

The right side at that point: about 16.31.

The same slope.

Right!

right side: 2xy ≈ 16.31

Let's look back.

First: separate x and y.

Then: integrate both sides.

Then: solve for y.

Then: use the starting value.

Last: check.

Answer