MathematicsChapter 4: Differential Equations

First-Order Linear Form

For y'+P(x)y=Q(x), integrating factor is e^(integral P dx).

Learning objectives

State the rule for first-order linear form.
Solve a first-order linear form problem step by step.
Check the result, restrictions, notation, and units.

Definition

For y'+P(x)y=Q(x), integrating factor is e^(integral P dx).

Mathematical form

y=Q(x)

y=e^x has factor e^x and (e^x y)

Where / symbols

First-order linear form: For y'+P(x)y=Q(x), integrating factor is e^(integral P dx).

CONCEPT AND RULE

For y'+P(x)y=Q(x), integrating factor is e^(integral P dx).

METHOD

For first-order linear form, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.

WORKED EXAMPLE

y'+y=e^x has factor e^x and (e^x y)'=e⁽2x).

COMMON MISTAKE

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

How to solve first-order linear form

Identify the mathematical objects, givens, restrictions, and required result.
Choose and write the relevant definition, theorem, or formula.
Substitute with brackets and carry out one justified step per line.
Simplify, state the final answer clearly, and verify it in the original setting.

Worked example

Worked example:
y'+y=e^x has factor e^x and (e^x y)'=e⁽2x).

In simple words

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

Key points

First-order linear form
rule
worked example

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.