Question:

If $sin\,x \frac{dy}{dx}+y\,cos\,x=x\,sinx$, then $\left(y-1\right)\,sin\,x=$

Updated On: Jul 6, 2022

$c-x\,sinx$
$c+x\,cosx$
$c-x\,cosx$
$c+x\,sinx$

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The Correct Option is C

Solution and Explanation

$ \frac{dy}{dx}+\left(sin\,x\cdot y\right)=x\,sin\,x$ $\Rightarrow y\,sin\,x=\int x\,sin\,x\,dx=-x\,cos\,x+sin\,x+c$ $\Rightarrow \left(y-1\right)\,sin\,x=c-x\,cos\,x$.

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Notes on Differential equations

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