Question

find the value of linear differential equation x. dy/dx+y=x cube.y (power six) ​

Answer 1

Step-by-step explanation:

Given--->

-----------

linear differential equation as follows

dy

x ---- + y =x³y^6

dx

To find--->

------------

solution of given differential equation

solution ---->

--------------

formulee used in solving -->

-----------------------------------------

1.

d d d

---------(u v) =u -----(v) + v -------(u)

dx dx dx

2.

solution of linear differential equation

dv

----- +p V = q

dx

where p and q are functions of x only

then

integration p integrating function = e

now solution is

v (I. F.)=integration {(I. F. ) q} dx + c

3.

logx^p

e = p

now returning to original problem

dy

x ----- + y =x ^3 y^6

dx

dividing whole equation by xy^6

x dy y x^3 y^6

----- ------ + --------- = -----------

x y^6 dx x y^6 x y^6

1 dy 1 1

-------- ------ + -------- -------- =x^2

y^6 dx x y^5

now let

1

-------- = v

y^5

y^(-5) = v

differentiating with respect to x

dy dv

-5 y^(-6) ----- = ------

dx dx

1 dy 1 dv

-------- ------- =- ------- -------

y^6 dx 5 dx

now putting in differential equation

1 dv 1

- ------- ------ + ------ v = x^2

5 dx x

multiplying by (-5) in whole equation

dv 5

------ +(- ------)v = -5x^2

dx x

5

here p=- ( ------) ,q =-5x^2

x

5

integration of (- -------)

x

I. F. =e

-5logx logx^(-5)

= e = e

=x^(-5)

now solution is

vx^(-5)= integration of (-5) x^2 x^(-5) dx

+c

1

--------------= integration (-5) x^(-3) dx + c

x^5y^5

x(-3+1)

= (-5) -------------- + c

(-3+1)

x^(-2)

=-5 --------------- + c

-2

1 5

------------ =------------ + c

x^5y^5 2x^2

hope it helps you

Thanks