Separable Growth and Decay

Separable Growth and Decay

What is it?

dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.

In simple words

• 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.

Important terms

Separable growth and decay: dy/dt=ky gives y=Ae^(kt); sign of k decides growth or decay.

What you should learn

• 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.

Full school-level explanation

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

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 growth and decay
• rule
• worked example
• worked example

How the ideas connect

Worked example: If P(0)=100 and k=.02, P=100e⁽.02t).

Deeper understanding

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

How the ideas connect

Worked example: If P(0)=100 and k=.02, P=100e⁽.02t).

Common mistakes

Exam focus

Core Idea

Rule / Formula

Method

Worked Example

Exam focus

Quick revision

• Separable growth and decay
• rule
• worked example
• 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 match

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 Growth and Decay: define clearly → explain the main parts → give one example → state one exam caution.