Questions
3–5 questions per JEE paper
Difficulty
Medium-Hard
Importance
High yield for JEE and BITSAT
Overview
Differential Equations serve as the mathematical foundation for modeling physical systems, from radioactive decay to mechanical oscillations. Mastering this topic is essential for JEE and competitive engineering exams because it tests both algebraic proficiency and conceptual integration of calculus. The core task is transforming word problems or complex expressions into integrable forms to solve for unknown functions.
Order and Degree
The order is defined by the highest derivative present, while the degree is the power of that highest derivative after making the equation a polynomial in derivatives. Aspirants must eliminate radicals or fractions from derivatives before determining the degree.
- Order: Highest derivative index
- Degree: Exponent of highest derivative
- Equation must be a polynomial in terms of y', y'', etc.
- Degree is undefined for transcendental functions like sin(dy/dx)
Variable Separable Form
This is the most fundamental technique where variables x and y are isolated on opposite sides of the equation. Once separated, integration of both sides yields the general solution, usually accompanied by an arbitrary constant C.
- Form: f(x)dx = g(y)dy
- Integration: integral of f(x) dx = integral of g(y) dy
- Always add constant C to the side with x
- Check for singular solutions often missed in division
Homogeneous Differential Equations
Equations are homogeneous if they can be expressed in the form dy/dx = f(y/x). The standard strategy is to substitute y = vx, which transforms the equation into a separable variable form in terms of v and x.
- Substitution: y = vx implies dy/dx = v + x(dv/dx)
- Must verify if f(kx, ky) = f(x, y)
- Integration usually results in logarithmic forms
- Always substitute v back to y/x at the final step
Linear Differential Equations
A linear DE is of the form dy/dx + Py = Q, where P and Q are constants or functions of x only. The solution relies on finding the Integrating Factor (IF) to condense the LHS into the derivative of a product.
- Integrating Factor: IF = e^(integral P dx)
- General solution: y * IF = integral(Q * IF dx) + C
- If form is dx/dy + Px = Q, use IF = e^(integral P dy)
- Look for hidden linear forms by rearranging terms
Formula Sheet
Order = n for d^n(y)/dx^n
dy/dx = f(y/x) -> v + x(dv/dx) = f(v)
IF = e^(integral P dx)
y * e^(integral P dx) = integral(Q * e^(integral P dx) dx) + C
d/dx(u*v) = u(dv/dx) + v(du/dx)
Separable form: integral(dy/g(y)) = integral(f(x)dx)
Exam Tip
Always check for the Integrating Factor (IF) in complicated looking linear-style equations before attempting substitution methods; it is the fastest route to the solution.
Common Mistakes
- Forgetting the constant of integration C, leading to incorrect final expressions.
- Attempting to find the degree of an equation involving transcendental functions of derivatives.
- Neglecting to substitute v back to y/x in homogeneous equation problems.
More Revision Notes
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