MathematicsChapter 4: Differential Equations
Separable Equations
Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
Learning objectives
- State the rule for separable equations.
- Solve a separable equations problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
Formula
dx=xy gives dy/y
Where / symbols
- 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 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|=xfrac22+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
- 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:
- dy/dx=xy gives dy/y=x dx, so ln|y|=xfrac22+C.
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
- Separable equations
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Separable Equations
Sticky Notes · Formula and Worked Revision
Core Idea
- Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
Rule / Formula
- Write all y terms with dy and x terms with dx, integrate both sides, then solve if required.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- dy/dx=xy gives dy/y=x dx, so ln|y|=xfrac22+C.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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|=xfrac22+C.
- Final check: domain, sign, notation, units, and reasonableness.
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