MathematicsChapter 4: Differential Equations
General and Particular Solutions
A general solution contains arbitrary constants; conditions determine a particular solution.
Learning objectives
- State the rule for general and particular solutions.
- Solve a general and particular solutions problem step by step.
- Check the result, restrictions, notation, and units.
Definition
A general solution contains arbitrary constants; conditions determine a particular solution.
Mathematical form
dx=2x gives y
y=x2+3.
Where / symbols
- General and particular solutions
- A general solution contains arbitrary constants; conditions determine a particular solution.
CONCEPT AND RULE
A general solution contains arbitrary constants; conditions determine a particular solution.
METHOD
For general and particular solutions, 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=2x gives y=x2+C; y(0)=3 gives y=x2+3.
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 general and particular solutions
- 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=2x gives y=x2+C; y(0)=3 gives y=x2+3.
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
- General and particular solutions
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.General and Particular Solutions
Sticky Notes · Formula and Worked Revision
Core Idea
- A general solution contains arbitrary constants; conditions determine a particular solution.
Rule / Formula
- A general solution contains arbitrary constants; conditions determine a particular solution.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- dy/dx=2x gives y=x2+C; y(0)=3 gives y=x2+3.
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
- A general solution contains arbitrary constants; conditions determine a particular solution.
- Worked pattern: dy/dx=2x gives y=x^2+C; y(0)=3 gives y=x^2+3.
- Final check: domain, sign, notation, units, and reasonableness.
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