The solution of differential equation $2x \frac{dy}{dx} - y = 3$ represents a family of
- circles
- straight lines
- ellipses
- parabola
The Correct Option is D
Solution and Explanation
As we have, \(2 x \frac{d y}{d x}=y+3 \)
\(\Rightarrow \frac{2}{y+3} d y=\frac{d x}{x} \)
integrating, \(2 \ln (y+3) \)
\(=\ln x+\ln c=\ln c x \)
\(\Rightarrow \ln (y+3)^{2}=\ln c x\)
\(\Rightarrow(y+3)^{2}=c x \)
which is a family of parabolas.
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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