Ch 06/Linear DE — Integrating Factor Method/H.K. Dass — Section 3.9, p.147

Linear DE — Integrating Factor Method

Standard form: dy/dx + P(x)y = Q(x). Multiply by integrating factor μ = e^(∫P dx).

IStandard Form & Integrating Factor

Standard form of a first-order linear DE:

where P and Q are functions of x only.

Why multiply by e^{∫P dx}?

Because the left side is almost — but not quite — a derivative of a product. Multiplying by makes:

So the equation becomes , which integrates directly.

Note

When finding the integrating factor, drop the constant of integration — you only need one particular , not all of them.

Example 1 (H.K. Dass §3.9)easy

Solve .

Show step-by-step solution+

Standard form — identify P and Q

Compute ∫P dx

Find I.F.

Multiply DE by μ = x

Integrate both sides

Divide by x

Example 2 (H.K. Dass Ex.10)medium

Solve .

Show step-by-step solution+

Divide by (x+1) to get standard form

P = -1/(x+1), Q = eˣ(x+1)

I.F.

Multiply through

Integrate

General solution

Example 3medium

Solve , with .

Show step-by-step solution+

P = 1, Q = eˣ

I.F.

Multiply through

Integrate

Divide by eˣ

Apply y(0) = 1

Particular solution

Example 4hard

Solve .

Show step-by-step solution+

P = tan x, Q = sec x

I.F.

Multiply through

Integrate

General solution

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