Solve dy/dx = 4y, y(0) = -2 Step by Step

Step by step

The question

Question: solve dy/dx = 4y, with y(0) = −2.

dydx = 4y

y(0) = −2 is the starting value: when x is 0, y is −2.

y(0) = −2

The right side splits into an x part, 4, times a y part, y.

That is a separable equation.

separable: 4 × y

1 Separate x and y

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

1y × dydx = 4

÷ y both sides

Now multiply both sides by dx.

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

1y dy = 4 dx

2 Integrate both sides

Put an integral sign on both sides.

∫ 1y dy = ∫ 4 dx

On the left: ln |y|.

On the right: 4x.

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

ln |y| = 4x + 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^(4x + C)

A plus in the power splits into a times: e^(4x + C) = e^C × e^(4x).

|y| = e^(C) × e^(4x)

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^(4x) + C ✗

e^C is just some number.

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

y = Ae^(4x)

A = ±e^(C) (any number)

4 Use the starting value

Put in x = 0 and y = −2.

The power is 0 there, and e⁰ = 1.

−2 = Ae^(0)

So A = −2.

A = −2

So y = −2e^(4x).

y = −2e^(4x)

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ≈ −109.196.

x = 1: y ≈ −109.196

The slope between two very close points: about −436.79.

slope: y(1.001) − y(0.999)0.002 ≈ −436.79

The right side at that point: about −436.79.

The same slope.

Right!

right side: 4y ≈ −436.79

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