Steply

Separable equations

Differential Equations Ā· 20 problems, easy to harder

ā–¶ Watch the first one solved

How we do it

  1. Separate x and y
  2. Integrate both sides
  3. Solve for y
  4. Use the starting value
  5. Check

The steps for Solve dy/dx = 2xy, y(0) = 3. Every solution names its steps, so you always know what comes next.

Worked examples

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

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 Ɨ y

1 Separate x and y

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

1y Ɨ dydx = 2x
Ć· 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.

āˆ« 1y dy = āˆ« 2x dx

On the left: ln |y|.

On the right: xĀ².

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.

āˆ« 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^(xĀ² + C)

A plus in the power splits into a times: e^(xĀ² + C) = e^C Ɨ e^(xĀ²).

|y| = e^(C) Ɨ e^(xĀ²)
e^(a + b) = e^(a) Ɨ e^(b)

Careful!

The C is not added at the end.

e^C is a number that multiplies.

Careful: not y = e^(xĀ²) + C āœ—

e^C is just some number.

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

y = Ae^(xĀ²)
A = Ā±e^(C) (any number)

4 Use the starting value

Put in x = 0 and y = 3.

The power is 0 there, and eā° = 1.

3 = Ae^(0)

So A = 3.

A = 3

So y = 3e^(xĀ²).

y = 3e^(xĀ²)

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ā‰ˆ 8.155.

x = 1: y ā‰ˆ 8.155

The slope between two very close points: about 16.31.

slope: y(1.001) āˆ’ y(0.999)0.002 ā‰ˆ 16.31

The right side at that point: about 16.31.

The same slope.

Right!

right side: 2xy ā‰ˆ 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
y = 3e^(xĀ²)

ā–¶ Watch it solved

Solve dy/dx = xy^2, y(0) = 1 Medium

The question

Question: solve dy/dx = xyĀ², with y(0) = 1.

dydx = xy2

y(0) = 1 is the starting value: when x is 0, y is 1.

y(0) = 1

The right side splits into an x part, x, times a y part, yĀ².

That is a separable equation.

separable: x Ɨ y2

1 Separate x and y

Divide both sides by yĀ², so y is only on the left.

1y2 Ɨ dydx = x
Ć· yĀ² both sides

Now multiply both sides by dx.

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

1y2 dy = x dx

2 Integrate both sides

Put an integral sign on both sides.

āˆ« 1y2 dy = āˆ« x dx

On the left: āˆ’1 over y.

On the right: Ā½xĀ².

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

āˆ’1y = 12x2 + C

The power rule backwards, for y.

āˆ« yāˆ’2 dy = yāˆ’1āˆ’1 = āˆ’1y

3 Use the starting value

Put in x = 0 and y = 1.

The x side gives 0.

āˆ’11 = 0 + C

So C = āˆ’1.

C = āˆ’1

4 Solve for y

Multiply both sides by y.

āˆ’1 = y(12x2 + āˆ’1)

Careful!

The C stays inside the bottom: it came with Ā½xĀ².

Careful: not y = āˆ’112x2 + C āœ—

Divide both sides by Ā½xĀ² āˆ’ 1.

y = āˆ’112x2 āˆ’ 1

No fraction inside a fraction: times the top and the bottom by 2.

y = āˆ’2x2 āˆ’ 2

So y = āˆ’2/(xĀ² āˆ’ 2).

y = āˆ’2x2 āˆ’ 2

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ā‰ˆ 2.

x = 1: y ā‰ˆ 2

The slope between two very close points: about 4.

slope: y(1.001) āˆ’ y(0.999)0.002 ā‰ˆ 4

The right side at that point: about 4.

The same slope.

Right!

right side: xyĀ² ā‰ˆ 4

Let's look back.

First: separate x and y.

Then: integrate both sides.

Then: use the starting value.

Then: solve for y.

Last: check.

Answer
y = āˆ’2x2 āˆ’ 2

ā–¶ Watch it solved

Solve dy/dx = x/y, y(2) = 3 Harder

The question

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

dydx = xy

y(2) = 3 is the starting value: when x is 2, y is 3.

y(2) = 3

The right side splits into an x part, x, times a y part, 1 over y.

That is a separable equation.

separable: x Ɨ 1y

1 Separate x and y

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

y Ɨ dydx = x
Ɨ y both sides

Now multiply both sides by dx.

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

y dy = x dx

2 Integrate both sides

Put an integral sign on both sides.

āˆ« y dy = āˆ« x dx

On the left: yĀ² over 2.

On the right: Ā½xĀ².

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

y22 = 12x2 + C

The power rule backwards, for y.

āˆ« y dy = y22

3 Use the starting value

Put in x = 2 and y = 3.

The x side gives 2.

322 = 2 + C

So C = 5/2.

C = 52

4 Solve for y

Multiply both sides by 2.

y2 = x2 + 5

Careful!

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

It stays as one root.

Careful: not āˆš(x2 + 5) = x + 5 āœ—

Take the square root.

y is positive, so we keep the plus root: y = āˆš(xĀ² + 5).

y = āˆšx2 + 5

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ā‰ˆ 2.449.

x = 1: y ā‰ˆ 2.449

The slope between two very close points: about 0.41.

slope: y(1.001) āˆ’ y(0.999)0.002 ā‰ˆ 0.41

The right side at that point: about 0.41.

The same slope.

Right!

right side: x/y ā‰ˆ 0.41

Let's look back.

First: separate x and y.

Then: integrate both sides.

Then: use the starting value.

Then: solve for y.

Last: check.

Answer
y = āˆšx2 + 5

ā–¶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. solve dy/dx = 2xy, y(0) = 3
    Answer
    y = 3e^(xĀ²)
  2. solve dy/dx = -2y, y(0) = 5
    Answer
    y = 5e^(āˆ’2x)
  3. solve dy/dx = x/y, y(0) = 3
    Answer
    y = āˆšx2 + 9
  4. solve dy/dx = 3x^2 y, y(0) = 4
    Answer
    y = 4e^(xĀ³)
  5. solve dy/dx = 4y, y(0) = -2
    Answer
    y = āˆ’2e^(4x)
  6. solve dy/dx = x/y, y(0) = -2
    Answer
    y = āˆ’āˆšx2 + 4
  7. solve dy/dx = 2x/y^2, y(0) = 2
    Answer
    y = 3āˆš3x2 + 8

Medium

  1. solve dy/dx = xy^2, y(0) = 1
    Answer
    y = āˆ’2x2 āˆ’ 2
  2. solve dy/dx = y^2, y(0) = 1
    Answer
    y = āˆ’1x āˆ’ 1
  3. solve dy/dx = x^2/y^2, y(0) = 2
    Answer
    y = 3āˆšx3 + 8
  4. solve dy/dx = -x/y, y(0) = 5
    Answer
    y = āˆšāˆ’x2 + 25
  5. solve dy/dx = 6x^2 y, y(0) = 2
    Answer
    y = 2e^(2xĀ³)
  6. solve dy/dx = (2x + 3)y, y(0) = 1
    Answer
    y = e^(xĀ² + 3x)
  7. solve dy/dx = 2x/y, y(1) = 2
    Answer
    y = āˆš2x2 + 2

Harder

  1. solve dy/dx = x/y, y(2) = 3
    Answer
    y = āˆšx2 + 5
  2. solve dy/dx = xy^2, y(0) = 2
    Answer
    y = āˆ’2x2 āˆ’ 1
  3. solve dy/dx = 3x^2/y^2, y(0) = 2
    Answer
    y = 3āˆš3x3 + 8
  4. solve dy/dx = -2xy, y(0) = 7
    Answer
    y = 7e^(āˆ’xĀ²)
  5. solve dy/dx = 2y^2, y(0) = -1
    Answer
    y = āˆ’12x + 1
  6. solve dy/dx = 4x^3/y, y(0) = 3
    Answer
    y = āˆš2x4 + 9