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Linear Differential Equations and Integrating Factor

A first-order linear differential equation in y has the form dy/dx + P(x)y = Q(x), where P and Q are functions of x. It is solved using the integrating factor e^(∫P(x) dx).

In one line: Linear form first, IF from P, then y times IF equals integral of Q times IF.

Practice This Concept

Understand it

The integrating factor changes the left side into the derivative of y multiplied by the integrating factor. After multiplying by IF, the equation becomes d/dx(y·IF) = Q·IF. Integrating both sides gives y·IF = ∫Q·IF dx + C.

Example

For dy/dx + y = ex, P = 1 and Q = ex. IF = e^(∫1 dx) = ex. Then y ex = ∫ex·ex dx + C = ∫e2x dx + C = e2x/2 + C.

Formula and rule

Standard form: dy/dx + P(x)y = Q(x).

Condition: y and dy/dx occur to the first power, and P, Q are functions of x only.

Integrating factor: IF = e^(∫P dx).

Solution rule: y·IF = ∫Q·IF dx + C.

For dx/dy + P(y)x = Q(y), use IF = e^(∫P(y) dy).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Example: Solve dy/dx + (1/x)y = x, x > 0.

  1. 1Here P = 1/x and Q = x.
  2. 2IF = e^(∫1/x dx) = e^(log x) = x.
  3. 3Multiply by IF: x(dy/dx) + y = x2.
  4. 4Since left side is d(xy)/dx, d(xy)/dx = x2.
  5. 5Integrate: xy = x3/3 + C.
  6. 6Hence y = x2/3 + C/x.

Method you can reuse

  1. 1Rewrite the equation so the coefficient of dy/dx is 1.
  2. 2Identify P(x) and Q(x).
  3. 3Find IF = e^(∫P dx).
  4. 4Multiply the whole equation by IF.
  5. 5Write the left side as d(y·IF)/dx.
  6. 6Integrate the right side Q·IF.
  7. 7Add C and solve for y if required.

Visual tip

A diagram is not required for standard linear differential equation solving. If a case-study question describes growth, decay, or motion, a table or graph may support interpretation but is not essential to the method.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the integrating factor of dy/dx + (2/x)y = x3, for x > 0?

Think first, then show answer
Here P = 2/x, so IF = e^(∫2/x dx) = e^(2 log x) = x2.

Common wrong answer

Taking P before dividing by the coefficient of dy/dx, or forgetting to multiply Q by the integrating factor before integration.

Don't mix this up

Students sometimes use e^(∫Q dx) as the integrating factor. The integrating factor depends on P, not Q, after the equation is in dy/dx + Py = Q form.

Exam tip

First divide by the coefficient of dy/dx so that its coefficient becomes 1. Only then identify P and Q and calculate the integrating factor.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often appears as a long-answer problem or as part of an application-based question. The equation may need rearrangement before it matches dy/dx + Py = Q.
  1. 1

    Enough for 1 mark

    A first-order linear differential equation in y has the form dy/dx + P(x)y = Q(x), where P and Q are functions of x. It is solved using the integrating factor e^(∫P(x) dx).

  2. 2

    Add for 2 marks

    Example: For dy/dx + y = ex, P = 1 and Q = ex. IF = e^(∫1 dx) = ex. Then y ex = ∫ex·ex dx + C = ∫e2x dx + C = e2x/2 + C.

  3. 3

    Add for 3 marks

    The integrating factor changes the left side into the derivative of y multiplied by the integrating factor.

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