If order and degree of the differential equation corresponding to the family of curves y2 = 4a(x+a)(a is parameter) are m and n respectively, then m+n2 =
If order and degree of the differential equation corresponding to the family of curves y2 = 4a(x+a)(a is parameter) are m and n respectively, then m+n2 =
3
4
5
2
The Correct Option is C
Solution and Explanation
We are given the family of curves:
\[ y^2 = 4a(x + a) \]
1. Differentiate with respect to \( x \):
\[ 2y \frac{dy}{dx} = 4a \Rightarrow a = \frac{y}{2}\frac{dy}{dx} \]
2. Substitute \( a \) into the original equation:
\[ y^2 = 4\left(\frac{y}{2}\frac{dy}{dx}\right)\left(x + \frac{y}{2}\frac{dy}{dx}\right) \]
Simplify:
\[ y^2 = 2y\frac{dy}{dx}\left(x + \frac{y}{2}\frac{dy}{dx}\right) \]
\[ y^2 = 2xy\frac{dy}{dx} + y^2\left(\frac{dy}{dx}\right)^2 \]
Divide by \( y^2 \) (assuming \( y \neq 0 \)):
\[ 1 = \frac{2x}{y}\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 \]
3. This is a differential equation involving \( \frac{dy}{dx} \).
Order \( m = 1 \) (highest derivative is first order).
Degree \( n = 2 \) (highest power of \( \frac{dy}{dx} \) is 2).
Thus,
\[ m + n^2 = 1 + 4 = 5 \]
Hence, the answer is \( 5 \).
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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