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# Solve the Differential Equation: (1 +X2 ) Dy + 2xy Dx = Cot X Dx - Mathematics

Sum

Solve the differential equation: (1 +x) dy + 2xy dx = cot x dx

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#### Solution

(1 +x) dy + 2xy dx = cot x dx

(1 +x) dy/dx  + 2xy = cot x

dy/dx +(2xy)/(1+x^2) = (cot x)/(1 +x^2)

dy/dx + py = q

Comparing with linear differential equation

I .F . = e^(intp  dx)

I.F = e^(int x/(1+x^2) dx)

I .F = int (2x)/(1+x^2) dx

put  t = 1 +x

dt/dx = 2x

dt = 2x dx

I = int dt/t = "In " t = "In" ( 1+ x^2 )

⇒ I .f = e^("in"^((1+x^2) ) = 1 +x^2

x(I.F) = int  Q ( I .F) dx + c

x(1 +x^2 ) = int (cot x ) /(1 +x^2 )  (1+ x^2 )  dx + c

 x(1 + x^2 ) = int  cot x  dx + c

x + x3 = In | sin x | + c

x + x3 - In | sin x | + c + 0

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