MathematicsChapter 4: Differential Equations
Separable Growth and Decay
dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
Learning objectives
- State the rule for separable growth and decay.
- Solve a separable growth and decay problem step by step.
- Check the result, restrictions, notation, and units.
Definition
dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
Mathematical form
dt=ky gives y
k=.02
P=100e(.02t).
Where / symbols
- Separable growth and decay
- dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
CONCEPT AND RULE
dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
METHOD
For separable growth and decay, 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
If P(0)=100 and k=.02, P=100e(.02t).
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 growth and decay
- 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:
- If P(0)=100 and k=.02, P=100e(.02t).
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 growth and decay
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Separable Growth and Decay
Sticky Notes · Formula and Worked Revision
Core Idea
- dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
Rule / Formula
- dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- If P(0)=100 and k=.02, P=100e(.02t).
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
- dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.
- Worked pattern: If P(0)=100 and k=.02, P=100e(.02t).
- Final check: domain, sign, notation, units, and reasonableness.
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