Question

Subject test

Integrating factor of the differential equation y+y^3/3+x^2/2)dx +1/4(x+xy^2)Dy=0 Integrating factor of the differential equation

(a) x

(b)x^2

(c) x^3

(d) x^4

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Answer 1

Answer:

Subject test

Integrating factor of the differential equation y+y^3/3+x^2/2)dx +1/4(x+xy^2)Dy=0 Integrating factor of the differential equation

(a) x

(b)x^2

(c) x^3

(d) x^4

Step-by-step explanation:

Subject test

Integrating factor of the differential equation y+y^3/3+x^2/2)dx +1/4(x+xy^2)Dy=0 Integrating factor of the differential equation

(a) x

(b)x^2

(c) x^3

(d) x^4

Answer 2

Concept:

To solve this question, we first need to recall the concept of Integrating factor of a differential equation .

Whenever the given differential equation is not exact . then we have to multiply it by some function of x and y to make it exact, that funcction is called integrating factor denoted by I.F.

Given:

The differential equation is:

$(y+\frac{y^{3}}{3}+\frac{x^{2}}{2})dx +\frac{1}{4}(x+xy^{2}) dy =0$

To find:

The integrating factor of the given differential equation.

Solution:

On comparing the given equation with the general form M(x,y)dx+N(x,y)dy=0

we have M(x,y) = $y+\frac{y^{3}}{3}+\frac{x^{2}}{2}$

and         N(x,y) = $\frac{1}{4}(x+xy^{2})$

                   $\frac{dM}{dy} =1+y^{2}\\ \\\;\frac{dN}{dx} = \frac{1}{4} (1+y^{2})$

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

Then the differential equation is not exact.

The function, f(x) = $\frac{1}{N} (\frac{dM}{dy}-\frac{dN}{dx})$

                            = $\frac{4}{x(1+y^{2})} (1+y^{2}-\frac{1}{4}-\frac{y^{2}}{4})$

                            = $\frac{4}{x(1+y^{2})} (\frac{3}{4}(1+y^{2}))$

                            =  $\frac{3}{x}$

Integrating factor = $e^{\int f(x)dx}$

                             = $e^{\int \frac{3}{x}dx}$

                             = $e^{3\;log \;x}$                          (using alog b= log $b^{a}$ )

                             = $e^{log \;x^{3}}$

                             =   $x^{3}$

Hence the Integrating factor is $x^{3}$.

Option (c) is correct choice.