Question

Which of the following equations is an exact differential equation?
a. (2x+1) dx- xy dy = 0
b. x dy +(3x-2y) dx = 0
c.2xydx+(2+ 2x)dy = 0
d. 2xy dy - ydx=0
​

Answer 1

Step-by-step explanation:

2xydx+(2+ 2x)dy = 0

that's the answer

Answer 2

Concept:

We first recall the concept of exact differential equation to solve thiis question.

An equation that involves  a dependent variable, an independent variable and the derivatives of dependent variable in terms of independent variable is called a Differential Equation.

A equation of the form M(x,y)dx+N(x,y)dy=0 is said to be exact differential equation if

                  $\frac{dM}{dy} =\frac{dN}{dx}$              (1)

To find :

The exact differential equation from the given options.

Solution:

In option (a) the equation is (2x+1)dx-xy dy=0

On comparing this equation with equation (1)

we get M(x,y)=2x+1 and N(x,y) = -xy

then

$\frac{dM}{dy} =\frac{d(2x+1)}{dy}\\\frac{dM}{dy} =0$

and

$\frac{dN}{dx} =\frac{d(-xy)}{dx} \\\frac{dN}{dx} =-y$

So,  $\frac{dM}{dy} \neq \frac{dN}{dx}$

Therefore , the given differential equation is not exact.

In option (b) the equation is  xdy+(3x-2y) dx=0

On comparing this equation with equation (1)

we get M(x,y)=3x-2y and N(x,y) = x

then

$\frac{dM}{dy} =\frac{d(3x-2y)}{dy}\\\frac{dM}{dy} =-2$

and

$\frac{dN}{dx} =\frac{d(x)}{dx} \\\frac{dN}{dx} =1$

So,  $\frac{dM}{dy} \neq \frac{dN}{dx}$

Therefore , the given differential equation is not exact.

In option (c) the equation is 2xydx+(2+2x) dy=0

On comparing this equation with equation (1)

we get M(x,y)=2xy and N(x,y) = 2+2x

then

$\frac{dM}{dy} =\frac{d(2xy)}{dy}\\\frac{dM}{dy} =2x$

and

$\frac{dN}{dx} =\frac{d(2+2x)}{dx} \\\frac{dN}{dx} =2$

So,  $\frac{dM}{dy} = \frac{dN}{dx}$

Therefore , the given differential equation is exact.

In option (d) the equation is  2xydy-ydx=0

On comparing this equation with equation (1)

we get M(x,y)=-y and N(x,y) = 2xy

then

$\frac{dM}{dy} =\frac{d(-y)}{dy}\\\frac{dM}{dy} =-1$

and

$\frac{dN}{dx} =\frac{d(2xy)}{dx} \\\frac{dN}{dx} =2y$

So,  $\frac{dM}{dy} \neq \frac{dN}{dx}$

Therefore , the given differential equation is not exact.

Hence, the correct answer is option C

the equation 2xydx+(2+2x)dy=0 is exact differential equation