Solve dy/dx = -2y, y(0) = 5 Step by Step

Step by step

The question

Question: solve dy/dx = −2y, with y(0) = 5.

dydx = −2y

y(0) = 5 is the starting value: when x is 0, y is 5.

y(0) = 5

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

That is a separable equation.

separable: −2 × y

1 Separate x and y

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

1y × dydx = −2

÷ y both sides

Now multiply both sides by dx.

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

1y dy = −2 dx

2 Integrate both sides

Put an integral sign on both sides.

∫ 1y dy = ∫ −2 dx

On the left: ln |y|.

On the right: −2x.

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

ln |y| = −2x + 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^(−2x + C)

A plus in the power splits into a times: e^(−2x + C) = e^C × e^(−2x).

|y| = e^(C) × e^(−2x)

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^(−2x) + C ✗

e^C is just some number.

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

y = Ae^(−2x)

A = ±e^(C) (any number)

4 Use the starting value

Put in x = 0 and y = 5.

The power is 0 there, and e⁰ = 1.

5 = Ae^(0)

So A = 5.

A = 5

So y = 5e^(−2x).

y = 5e^(−2x)

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ≈ 0.677.

x = 1: y ≈ 0.677

The slope between two very close points: about −1.35.

slope: y(1.001) − y(0.999)0.002 ≈ −1.35

The right side at that point: about −1.35.

The same slope.

Right!

right side: −2y ≈ −1.35

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