Solve the differential equation ` x (dy )/(dx) - y = 2 x^(3), x gt 0`

1.	Solve the differential equation ` x (dy )/(dx) - y = 2 x^(3), x gt 0`
Answer» The given differential equation may be written as ` (dy)/(dx) + (( - 1 )/( x )) y = 2x ^(2)" " ` ... (i) This is of the form, ` (dy)/(dx) + Py = Q`, where `P = (-1)/(x) and Q = 2x ^(2)` Thus, the given diffential equation is linear. IF ` = e ^( int Pdx) = e ^( int - (1)/(x) dx ) = e ^(-log x ) = e ^( log ((1)/(x))) = (1)/(x)` So, the required solution is given by ` y xx IF = int {Q xx IF} dx + C` , where C is an arbitrary constant, i.e, ` y xx (1)/(x) = int ( 2x ^(2) xx (1)/(x)) dx + C = 2 int x dx + C = x ^(2) + C ` `rArr y = x ^(3) + Cx ` Hence, the required solution is `y = x ^(3) + Cx`

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