Questions
3–5 questions per paper
Difficulty
Medium
Importance
Core — never skip
Overview
Differential Equations form a cornerstone of calculus, describing the relationship between a function and its derivatives. For board exams, mastering the classification and systematic solving methods is essential as this topic carries significant weight and predictable question patterns.
Order and Degree of Differential Equations
The order is defined by the highest derivative present, while the degree is the power of that highest derivative after the equation is expressed as a polynomial in derivatives. Ensure all radicals and fractions are cleared from derivative terms before determining these values.
- Order: Highest derivative order present
- Degree: Highest power of the highest order derivative
- Equation must be a polynomial in terms of derivatives
- Degree is not defined if the equation contains transcendental functions like sin(dy/dx) or e^(d^2y/dx^2)
Variable Separable Method
This is the simplest method for solving first-order differential equations where variables x and y can be completely isolated on opposite sides of the equality. Once separated, simple integration on both sides yields the general solution.
- Form: f(x)dx = g(y)dy
- General solution: Integral of f(x) dx = Integral of g(y) dy + C
- Most common method for basic board exam problems
- Always add the constant of integration C
Linear Differential Equations
First-order linear differential equations are characterized by the form dy/dx + Py = Q, where P and Q are constants or functions of x only. Solving these relies on the calculation of the Integrating Factor (IF).
- Standard form: dy/dx + Py = Q
- Integrating Factor (IF) = e^(integral P dx)
- Solution: y * IF = integral (Q * IF) dx + C
- Dual case: dx/dy + Px = Q (where P, Q are functions of y)
- IF for dual case = e^(integral P dy)
Formula Sheet
Order = n for d^ny/dx^n
dy/dx = f(x)g(y) implies integral (1/g(y)) dy = integral f(x) dx
IF = e^(integral P dx)
y * e^(integral P dx) = integral (Q * e^(integral P dx)) dx + C
IF = e^(integral P dy) for dx/dy + Px = Q
Exam Tip
Always verify if the equation is a polynomial in derivatives before attempting to define the degree; if a derivative is inside a sine or exponential function, the degree is automatically undefined.
Common Mistakes
- Failing to define the degree as undefined when the derivative is inside a transcendental function.
- Forgetting the constant of integration C, leading to loss of marks in the general solution.
- Incorrectly identifying P and Q in the Linear Differential Equation form.
More Revision Notes
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