The differential equation dy/dx=√1-y2/y determines a family of circles with
The differential equation dy/dx=√1-y2/y determines a family of circles with
(A) Variable radius and fixed centre at (0,1)
(B) Variable radius and fixed centere at (0,-1)
(C) Fixed radius of 1 Unit and variable centre along the X-axis
(D) Fixed radius of 1 Unit and variable centre along the X- axis
The Correct Option is D
Solution and Explanation
The given differential equation is:
dy/dx = √(1-y^2)/y
We can write this equation in the form: dy/√(1-y^2) = dx/y
Integrating both sides: arcsin(y) = ln|x| + C Where C is the constant of integration. Solving for y: y = sin(ln|x| + C) This is the general solution of the differential equation.
We can observe that this solution represents a family of curves which are circles centered on the x-axis.
To see this, we can rewrite the solution as: y = sin(ln|x| + C) = (e^(ln|x|+C) - e^-(ln|x|+C))/2 Simplifying: y = (x - 1/x)/2
This is the equation of a circle centered at (0,0) with radius 1/2.
Therefore, the correct option is (D) fixed radius of 1 unit and variable center along the x-axis.
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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