hey ...........look

 

divide the whole equaion by y⁶

ok.......

 

we get

 

(y^-6)dy/dx    +  (y^-5 )/x  =   x²

 

now

 

 

put  y^-5  =t

differentiating we get

 

-5y^-6 dy = dt

thus writing the equation in  t form .............we  get

 

(-1/5) dt/ dx    +   t/ x     =    x²

 

now

 

 

now multipying the equaion by -5 we get

 

dt/dx   -   5 t/ x    =   -5x²

 

 

thus the equation has been converted into bernoulli form............

 

which says .....

 

dt/dx  +  Pt  =  Q

 

where P and Q should be only in form of x..............

 

therefore

 we get

 

the solution.....

 

t(IF) = _{[ ]}^{[ ]} Q( IF) dx

 

 

which is e  to the power of _{[ ]}^{[ ]} Pdx

 

 

here P = -5/x

 

thus IF= x^-5

 

thus by bernoulli form.......we get

 

t x^-5=  _{[ ]}^{[ ]} -5x²(x^-5)dx

 

we  get

 

tx^-5 =  5x^-2/2     +   c

 

now putting t= y^-5

 

 

we get the solution

 

 

 

i think this might be the answer.........

 

if no calculation error

 

 

 

don