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Junior Executive (ATC) Official Paper 2: Held on Nov 2018 - Shift 2

Option 3 : first order and second degree

Junior Executive (ATC) Official Paper 1: Held on Nov 2018 - Shift 1

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__Explanation:__

__\(x^2 \left(\dfrac{\partial z}{\partial x}\right)^2 = Z\left(Z - y \dfrac{\partial z}{\partial y}\right)\)__

The above equation is **first order and second degree**

Order of Differential Equation:

- Differential Equations are classified on the basis of the order.
- The order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.

Ex: \(\frac{{d^4y}}{{dx^4}}-\frac{{d^3y}}{{dx^3}}-9\frac{{dy}}{{dx}}+y=12\)

- In this equation, the order of the highest derivative is 4 hence this is a fourth-order differential equation.

Ex: \(4(\frac{{d^3y}}{{dx^3}})^2-\frac{{dy}}{{dx}}=12\)

- This equation represents a Third-order differential equation.
- This way we can have higher-order differential equations i.e. nth order differential equations.

Degree of Differential Equation:

- The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation.
- The differential equation must be a polynomial equation in derivatives for the degree to be defined.

Ex: \({\left[ {5 + {{\left( {\frac{{dy}}{{dx}}} \right)}^3}} \right]^2} = \frac{{{d^2}y}}{{d{x^2}}}\)

- Here, the exponent of the highest order derivative is one (i.e. Second-order) and the given differential equation is a polynomial equation in derivatives. Hence, the degree of this equation is 1.

Ex: \([\frac{{d^2y}}{{dx^2}}-9(\frac{{dy}}{{dx}})^2]^4=k^2(\frac{{d^3y}}{{dx^3}})^2\)

- The order of this equation is 3 and the degree is 2 as the highest derivative is of order 3 and the exponent raised to the highest derivative is 2.