IIT JEE Main Maths -Unit 9- Order and degree of differential equations- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 9- Order and degree of differential equations – Study Notes – New syllabus
IIT JEE Main Maths -Unit 9- Order and degree of differential equations – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Differential Equations — Definition, Order, and Degree
Differential Equations — Definition, Order, and Degree
A Differential Equation is an equation that involves an unknown function and its derivatives with respect to one or more independent variables.
In calculus and physics, differential equations describe how a quantity changes — for example, motion, growth, or decay.
Definition
A differential equation is an equation that contains one or more derivatives of an unknown function.
Example: \( \dfrac{dy}{dx} + y = 0 \)
Here, \( y \) is the dependent variable, and \( x \) is the independent variable.
Order of a Differential Equation
The order of a differential equation is the highest order of derivative present in the equation.
- Example: \( \dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} + y = 0 \)
- Order = 2 (because the highest derivative is \( \dfrac{d^2y}{dx^2} \))
Degree of a Differential Equation
The degree of a differential equation is the power of the highest order derivative, provided the equation is free from radicals and fractions involving derivatives.
- Example: \( \left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = 0 \)
- Order = 2, Degree = 3.
Conditions for Degree to Exist
The degree of a differential equation is defined only if:
- The equation is a polynomial in all derivatives of \( y \).
- There are no radicals, fractional, or negative powers of derivatives.
Examples
| Equation | Order | Degree |
|---|---|---|
| \( \dfrac{dy}{dx} + y = 0 \) | 1 | 1 |
| \( \left(\dfrac{d^2y}{dx^2}\right)^3 + 5\dfrac{dy}{dx} = 0 \) | 2 | 3 |
| \( \sqrt{\dfrac{dy}{dx}} + y = x \) | 1 | Not defined (derivative under radical) |
| \( \left( \dfrac{d^3y}{dx^3} \right)^2 + \dfrac{d^2y}{dx^2} = 0 \) | 3 | 2 |
Example
Find the order and degree of \( \dfrac{d^3y}{dx^3} + 2\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} + y = 0 \).
▶️ Answer / Explanation
The highest order derivative is \( \dfrac{d^3y}{dx^3} \) → Order = 3.
The power of the highest order derivative is 1 → Degree = 1.
Answer: Order = 3, Degree = 1.
Example
Find the order and degree of \( \left(\dfrac{d^2y}{dx^2}\right)^4 + 5\left(\dfrac{dy}{dx}\right)^3 + y = 0 \).
▶️ Answer / Explanation
The highest order derivative is \( \dfrac{d^2y}{dx^2} \) → Order = 2.
The power of this term is 4 → Degree = 4.
Answer: Order = 2, Degree = 4.
Example
Find the order and degree of \( \sqrt{\dfrac{d^2y}{dx^2}} + \dfrac{dy}{dx} = x \).
▶️ Answer / Explanation
The equation contains \( \sqrt{\dfrac{d^2y}{dx^2}} \), which is a radical in a derivative.
Therefore, the equation is not polynomial in derivatives.
Hence, degree is not defined.
However, the highest order derivative is \( \dfrac{d^2y}{dx^2} \) → Order = 2.
Answer: Order = 2, Degree = Not defined.
Notes and Study Materials
- Concepts of Differential Equations
- Differential Equations Master File
- Differential Equations Revision Notes
- Differential Equations Formulae
- Differential Equations Reference Book
- Differential Equations Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Differential Equations” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Ordinary differential equations, their order, and degree.Formation of differential equations. The solution of differential equations by the method of separation of variables. The solution of homogeneous and linear differential equations of the type
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