Solve dy/dx = e^(2x)/y Step by Step

Step by step

The question

Question: solve dy/dx = e^(2x) over y.

dydx = e^(2x)y

The right side splits into an x part, e^(2x), times a y part, 1 over y.

That is a separable equation.

separable: e^(2x) × 1y

1 Separate x and y

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

y × dydx = e^(2x)

× y both sides

Now multiply both sides by dx.

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

y dy = e^(2x) dx

2 Integrate both sides

Put an integral sign on both sides.

∫ y dy = ∫ e^(2x) dx

On the left: y² over 2.

On the right: ½e^(2x).

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

y²2 = 12e^(2x) + C

The power rule backwards, for y.

∫ y dy = y²2

3 Solve for y

Multiply both sides by 2.

y² = e^(2x) + 2C

2C is still just some number.

Call it C again.

y² = e^(2x) + C

2C = a new C

Careful!

The square root of a sum is not the sum of the roots.

It stays as one root.

Careful: not y = ±(e^(2x) + √C) ✗

Take the square root: plus or minus.

So y = ±√(e^(2x) + C).

y = ±√e^(2x) + C

4 Check

Let's check with a calculator.

Take C as 1, any number works.

At x = 1, our answer gives y ≈ 2.896.

C = 1, x = 1: y ≈ 2.896

The slope between two very close points: about 2.55.

slope: y(1.001) − y(0.999)0.002 ≈ 2.55

The right side at that point: about 2.55.

The same slope.

Right!

right side: e^(2x)/y ≈ 2.55

Let's look back.

First: separate x and y.

Then: integrate both sides.

Then: solve for y.

Last: check.

Answer