Chapter 9 – Differential Equations (JEE)
1. What is a Differential Equation?
A differential equation is an equation involving an independent variable,
a dependent variable and one or more derivatives of the dependent variable.
Example:
$\dfrac{dy}{dx} = 3x^2$
2. Order and Degree of a Differential Equation
Order is the order of the highest derivative present.
Degree is the power of the highest order derivative after removing radicals and fractions.
$\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^2 = x$
Order = 2, Degree = 1
Order = 2, Degree = 1
3. Types of Differential Equations
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Non-linear Differential Equations
For JEE, we deal mainly with first order ordinary differential equations.
4. General and Particular Solution
General solution contains arbitrary constant(s).
Particular solution is obtained by using given conditions.
$\dfrac{dy}{dx} = 2x \Rightarrow y = x^2 + C$
5. Formation of Differential Equation
Differential equation can be formed by eliminating arbitrary constants
from a given relation.
Number of constants eliminated = order of the differential equation.
6. Variable Separable Differential Equations
If a differential equation can be written as:
$f(y)dy = g(x)dx$
Then integrate both sides separately.
7. Homogeneous Differential Equations
A differential equation of the form:
$\dfrac{dy}{dx} = F\left(\dfrac{y}{x}\right)$
Use substitution $y = vx$.
8. Linear Differential Equations (First Order)
$\dfrac{dy}{dx} + Py = Q$
Where $P$ and $Q$ are functions of $x$ or constants.
9. Integrating Factor (I.F.)
$I.F. = e^{\int P dx}$
Solution:
$$y \cdot I.F. = \int (Q \cdot I.F.) dx + C$$
10. Exact Differential Equations
An equation of the form:
$Mdx + Ndy = 0$
is exact if:
$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$
11. Applications of Differential Equations
- Growth and decay problems
- Motion problems
- Population models
- Radioactive decay
12. Important Standard Results
$\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$
$\int \dfrac{1}{x} dx = \ln|x| + C$
13. Initial Value Problems
If a condition like $y=y_0$ at $x=x_0$ is given,
it is used to find constant $C$.
14. Common JEE Mistakes
- Forgetting constant of integration
- Wrong integrating factor
- Incorrect separation of variables
- Not checking exactness condition
15. Typical JEE Question Patterns
| Type | Method |
|---|---|
| Variable separable | Separate and integrate |
| Homogeneous | Substitute $y=vx$ |
| Linear | Use integrating factor |
| Initial value | Find constant using condition |
16. Final Revision Checklist
You have mastered Differential Equations if you can:
- Identify type of differential equation quickly
- Choose correct solving method
- Apply integrating factor accurately
- Handle initial value problems
- Avoid common algebraic mistakes