Linear First-Order Equations

Step 1: Let's Learn

Read it, or press Listen and follow the words.

A linear first-order equation can be written as dy/dx + P(x)y = Q(x). The unknown y and its derivative appear only to the first power.

Usually not separable

dy/dx + 2y = x cannot be pulled apart. A different method is needed.

The integrating factor

Multiply through by e raised to the integral of P. That single move makes the left side a perfect product-rule derivative.

Why it works

The left side becomes the derivative of the factor times y. Integrating both sides then removes the derivative entirely.

The procedure

Put it in standard form, compute the factor, multiply, recognise the product rule, integrate, then divide back out.

What the answer looks like

The solution splits into a part that fades and a part that persists. The fading part carries the arbitrary constant.

The standard form

A linear first-order equation is dy/dx + P(x)y = Q(x). Putting it in this form first is essential, because the integrating factor is computed from P and nothing else.

Multiply by something clever

Multiplying by e^(∫P dx) makes the left side an exact derivative of a product. That is the whole trick: the equation becomes directly integrable rather than merely rearrangeable.

Why that factor works

It is chosen precisely so the product rule reproduces the left-hand side. Working through that once explains why the exponential of an integral appears rather than something simpler.

When separation fails

Many linear equations do not separate. The integrating factor handles every linear first-order equation, which makes it the more general of the two techniques.

Step 2: Try It Yourself

Tap and try it out.

This field belongs to a linear equation that does not separate. Follow solutions from several starts and see how they converge.

The equationdy/dx = a·(x + y)
Through(-2, 2)

Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.

starting x

starting y

Step 3: Watch an Example

One step at a time.

Watch Sana Use an Integrating Factor

Sana solves dy/dx + 2y = 6.

Step 1
She reads P = 2 from the standard form.

Step 4: Your Turn

Practice makes it stick.

The Factor

Problem 1 of 2

dy/dx + 5y = 1. What is the coefficient P?

The Long Run

Problem 2 of 2

y = 3 + Ce^(−2x). What value does y approach as x grows large?

Multiply Through

1 of 8

dy/dx + 4y = 8. What is P?

2 of 8

dy/dx + 4y = 8. What constant value does y settle toward?

3 of 8

dy/dx + 3y = 12. What constant value does y settle toward?

4 of 8

Is dy/dx + 2y = x linear? 1 yes, 0 no.

5 of 8

Is dy/dx + y² = x linear? 1 yes, 0 no.

6 of 8

y = 7 + Ce^(−3x) with y = 10 at x = 0. What is C?

7 of 8

Order the steps of the integrating factor method.

1Compute the integrating factor from P
2Multiply through and recognise the product rule
3Integrate both sides and divide the factor back out
4Write the equation in standard form

8 of 8

y = 5 + Ce^(−x). What value does y approach as x grows large?

Step 5: Quick Check

Show what you know.

Question 1 of 2

dy/dx + 6y = 18. What constant value does y settle toward?

Question 2 of 2

What does the integrating factor accomplish?

What You Learned

A linear first-order equation has the form dy/dx + P(x)y = Q(x).
Multiplying by e to the integral of P makes the left side a product-rule derivative.
The solution splits into a fading part carrying the constant and a persisting part.