Separable Equations

Separable Equations

What is it?

Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.

In simple words

• Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
• Worked pattern: dy/dx=xy gives dy/y=x dx, so ln|y|=x^frac22+C.
• Final check: domain, sign, notation, units, and reasonableness.

Important terms

Separable equations: Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.

What you should learn

• State the rule for separable equations.
• Solve a separable equations problem step by step.
• Check the result, restrictions, notation, and units.

Full school-level explanation

Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.

CONCEPT AND RULE

Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.

METHOD

For separable equations, 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

dy/dx=xy gives dy/y=x dx, so ln|y|=x^frac22+C.

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 separable equations

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

• Separable equations
• rule
• worked example
• worked example

How the ideas connect

Worked example: dy/dx=xy gives dy/y=x dx, so ln|y|=x^frac22+C.

Deeper understanding

Go beyond memorising Separable Equations. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked example: dy/dx=xy gives dy/y=x dx, so ln|y|=x^frac22+C.

Common mistakes

Exam focus

Core Idea

Rule / Formula

Method

Worked Example

Exam focus

Quick revision

• Separable equations
• rule
• worked example
• Separable equations: Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
• CONCEPT AND RULE: Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
• METHOD: For separable equations, identify the given information and required result. Select the matching def

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 Separable Equations: define clearly → explain the main parts → give one example → state one exam caution.