First-order Linear Form

First-order Linear Form

What is it?

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

In simple words

• For y'+P(x)y=Q(x), integrating factor is e^(integral P dx).
• Worked pattern: y'+y=e^x has factor e^x and (e^x y)'=e^(2x).
• Final check: domain, sign, notation, units, and reasonableness.

Important terms

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

What you should learn

• 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.

Full school-level explanation

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

How it works

• 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.

Remember

• First-order linear form
• rule
• worked example
• worked example

How the ideas connect

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

Deeper understanding

Go beyond memorising First-order Linear Form. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

Rule / Formula

Method

Worked Example

Exam focus

Quick revision

• First-order linear form
• rule
• worked example
• 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

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

For long answers on First-order Linear Form: define clearly → explain the main parts → give one example → state one exam caution.