DIFFERENTIAL EQUATIONS
Complete Exam Revision Notes
1. BASIC CONCEPTS
• A differential equation is an equation involving a dependent variable
and its derivatives.
• Order = highest order derivative present in the equation.
• Degree = power of the highest order derivative, after the equation is
made polynomial in derivatives.
Examples:
dy/dx + y = x → Order = 1, Degree = 1
d²y/dx² + 3dy/dx + y = 0 → Order = 2, Degree = 1
GENERAL & PARTICULAR SOLUTION
General solution contains arbitrary constants.
Example: y = Cx + C₁ contains arbitrary constants.
Particular solution is obtained after using given conditions to find
constants.
KEY IDEA
Always identify the type of differential equation first. The method
depends on its form.
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, FORMATION OF DIFFERENTIAL EQUATIONS
2. FORMATION BY ELIMINATING ARBITRARY CONSTANTS
Steps:
1. Write the given equation.
2. Differentiate it as many times as the number of arbitrary constants.
3. Eliminate the constants.
Example:
y = Cx²
Differentiate: dy/dx = 2Cx
Since C = y/x²,
dy/dx = 2y/x
Therefore: x(dy/dx) − 2y = 0
Another example:
y = C₁ cos x + C₂ sin x
Differentiate twice and eliminate C₁, C₂.
Result: d²y/dx² + y = 0
REMEMBER
Number of arbitrary constants generally tells you how many times
differentiation is required.
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Complete Exam Revision Notes
1. BASIC CONCEPTS
• A differential equation is an equation involving a dependent variable
and its derivatives.
• Order = highest order derivative present in the equation.
• Degree = power of the highest order derivative, after the equation is
made polynomial in derivatives.
Examples:
dy/dx + y = x → Order = 1, Degree = 1
d²y/dx² + 3dy/dx + y = 0 → Order = 2, Degree = 1
GENERAL & PARTICULAR SOLUTION
General solution contains arbitrary constants.
Example: y = Cx + C₁ contains arbitrary constants.
Particular solution is obtained after using given conditions to find
constants.
KEY IDEA
Always identify the type of differential equation first. The method
depends on its form.
Page
, FORMATION OF DIFFERENTIAL EQUATIONS
2. FORMATION BY ELIMINATING ARBITRARY CONSTANTS
Steps:
1. Write the given equation.
2. Differentiate it as many times as the number of arbitrary constants.
3. Eliminate the constants.
Example:
y = Cx²
Differentiate: dy/dx = 2Cx
Since C = y/x²,
dy/dx = 2y/x
Therefore: x(dy/dx) − 2y = 0
Another example:
y = C₁ cos x + C₂ sin x
Differentiate twice and eliminate C₁, C₂.
Result: d²y/dx² + y = 0
REMEMBER
Number of arbitrary constants generally tells you how many times
differentiation is required.
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