Find the general solution of the differential equation dy/dx + y.cotx

1.	Find the general solution of the differential equation dy/dx + y.cotx = 2x + x2.cotx
Answer» dy/dx + y.cot x = 2x + x² .cotx Let P = cot x, Q = 2x + x²cot x I.F = e^∫pdx = e^∫cotxdx = e^{∫log(sin x)} = sin x Sol is y (I.F) = ∫ (QxI.F) dx +C y sinx = ∫(2x + x²cotx)sin dx + c = ∫(2x + x² cotx) dx +c = 2 ∫ x sin xdx + ∫x² cosx dx + c = 2[x(-cosx)-(1)(-sinx)] +[x² sinx - 2x(-cosx) + 2c(-sinx)] =2xcos x + 2sin x + x²sinx y sin x = x² sinx + c This is the Required General solution for the given differential Equation.

Find the general solution of the differential equation dy/dx + y.cotx = 2x + x2.cotx

Answer»

dy/dx + y.cot x = 2x + x² .cotx

Let P = cot x, Q = 2x + x²cot x

I.F = e^∫pdx = e^∫cotxdx = e^{∫log(sin x)} = sin x

Sol is

y (I.F) = ∫ (QxI.F) dx +C

y sinx = ∫(2x + x²cotx)sin dx + c

= ∫(2x + x² cotx) dx +c

= 2 ∫ x sin xdx + ∫x² cosx dx + c

= 2[x(-cosx)-(1)(-sinx)] +[x² sinx - 2x(-cosx) + 2c(-sinx)] =2xcos x + 2sin x + x²sinx y sin x

= x² sinx + c

This is the Required General solution for the given differential Equation.

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Find the general solution of the differential equation dy/dx + y.cotx = 2x + x2.cotx

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