Question:
The solution for the differential equation ${ \frac{dy}{y} +\frac{ dx}{x} = 0 }$ is
The solution for the differential equation ${ \frac{dy}{y} +\frac{ dx}{x} = 0 }$ is
Updated On: Apr 14, 2024
- ${\frac{1}{y} + \frac{1}{x} = c }$
- $\log \, x . \log \, y = c$
- $x \, y = c$
- $x + y = c$
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The Correct Option is C
Solution and Explanation
We have, $\frac{d y}{y}+\frac{d x}{x}=0$
On integrating both sides, we get
$\log\, y+\log \,x=\log \,c$
$\Rightarrow \log \,x y=\log \,c$
$x y=c$
On integrating both sides, we get
$\log\, y+\log \,x=\log \,c$
$\Rightarrow \log \,x y=\log \,c$
$x y=c$
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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