For the differential equation [1 + \((\frac {dy}{dx})^2\)]5/2 = 8 \((\frac {d^2y}{dx^2})\) has the order and degree_________respectively.
For the differential equation [1 + \((\frac {dy}{dx})^2\)]5/2 = 8 \((\frac {d^2y}{dx^2})\) has the order and degree_________respectively.
- 2 and 6
- 2 and 3
- 2 and 2
- 2 and 1
The Correct Option is C
Solution and Explanation
The given differential equation is [1 + \((\frac {dy}{dx})^2\)]5/2 = 8\((\frac {d^2y}{dx^2})\).
The highest order derivative in the equation is the second derivative, \((\frac {d^2y}{dx^2})\). Therefore, the order of the differential equation is 2.
Next, we need to determine the degree of the differential equation, which is the highest power of the derivative. In this case, the second derivative appears inside the square root [1 + \((\frac {dy}{dx})^2\)]5/2. The highest power of the derivative is 2 raised to the power of \((\frac {5}{2})\), which is (25/2). Therefore, the degree of the differential equation is 2.
Hence, the order and degree of the given differential equation are 2 and 2, respectively.
The correct option is (C) 2 and 2.
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations