1 1 point Find the general form of the following differentia

1. [1 point] Find the general form of the following differential equation by using the method of integrating factors. (Note: You should review some of your basic trigonometric identities.) 2sin^2 x? y\' + sin(2x)?y = 6x^2 sin x

Solution

Given differential equation: 2 sin^2x.y\'+ sin(2x).y=6 x^2 sinx

if solve this: y\'+ sin(2x)/2 sin^2x..y=(6 x^2 sinx/2 sin^2x.)

divid every term with 2 sin^2x.

y\'+ 2 sinx.cosx/2 sin^2x.y=(6 x^2 sinx/2 sin^2x.) [sin2x=2sinx.cosx]

y\'+ 2 sinx.cosx/2 sinxsinx.y=(6 x^2 sinx/2 sinxsinx.) [2sin^2x=2sinx.sinx]

y\'+ cotx.y=3 x^2 /sinx. [cos/sin=cot] ------------------(1)

y\'+ cotx.y=3 x^2 coscex . [cos/sin=cot]

this is like dy/dx+P(x).y=Q(x) standard form

After writing the equation in standard form, P(x) can be identified. One then multiplies the equation by the following “integrating factor”: IF=e^(\\intP(x) dx)

IF=e^log|sinx|

IF=sinx

multiply 1 with IF

sinx y\'+sinx cotx.y=3 x^2 sinx/sinx. and integrating

  sinx y\'+cos x.y=3 x^2.  

general form is    y=3x^3-ylog|sin x|+c