MathematicsChapter 4: Differential Equations
Initial-Value Problems
Use the initial condition after integration to find constants without losing domain restrictions.
Learning objectives
- State the rule for initial-value problems.
- Solve a initial-value problems problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Use the initial condition after integration to find constants without losing domain restrictions.
Mathematical form
dx=3x2
y=x3+4.
Where / symbols
- Initial-value problems
- Use the initial condition after integration to find constants without losing domain restrictions.
CONCEPT AND RULE
Use the initial condition after integration to find constants without losing domain restrictions.
METHOD
For initial-value problems, 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=3x2, y(1)=5 gives y=x3+4.
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 initial-value problems
- 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=3x2, y(1)=5 gives y=x3+4.
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
- Initial-value problems
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Initial-Value Problems
Sticky Notes · Formula and Worked Revision
Core Idea
- Use the initial condition after integration to find constants without losing domain restrictions.
Rule / Formula
- Use the initial condition after integration to find constants without losing domain restrictions.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- dy/dx=3x2, y(1)=5 gives y=x3+4.
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
- Use the initial condition after integration to find constants without losing domain restrictions.
- Worked pattern: dy/dx=3x^2, y(1)=5 gives y=x^3+4.
- Final check: domain, sign, notation, units, and reasonableness.
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