Question:
The solution of the differential equation $xdy + ydx = xydx$ when $y(1) = 1$ is
The solution of the differential equation $xdy + ydx = xydx$ when $y(1) = 1$ is
Updated On: Jul 7, 2022
- $y=\frac{e^{x}}{x}$
- $\frac{e^{x}}{ex}$
- $y=\frac{xe^{x}}{e}$
- none of these
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The Correct Option is B
Solution and Explanation
From given $\frac{xdy+ydx}{xy}=dx$
On integration, we get
$log_{e}\left(xy\right)=x+k$
$\therefore x=1$,
$y=1$
$\Rightarrow k=-1$
$\Rightarrow xy=e^{x-1}$
$\Rightarrow xy=\frac{e^{x}}{e}$
$\therefore y=\frac{e^{x}}{ex}$
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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