Solve the following initial value differential equation (x − 1) dy/dx = 2xy, when y(2) = 1

 Solve the following initial value differential equation

(x − 1) dy/dx = 2xy, when y(2) = 1.

This is a variable  separable differential equation.

 Separate variables

(x − 1) dy/dx = 2xy

dy/y = [2x / (x − 1)] dx

 Integrate both sides

∫ dy/y = ∫ [2x / (x − 1)] dx

 2x/(x−1) = 2 + 2/(x−1) using long division or manipulation of the numerator

∫ dy/y =∫ [2 + 2/(x − 1)] dx 

ln|y| = 2x + 2ln|x − 1| + C

Apply initial condition y(2) = 1

When x = 2, y = 1

ln|1| = 2(2) + 2ln|2 − 1| + C

0 = 4 + 2ln(1) + C

0 = 4 + 0 + C

  ⇒  C = −4

ln|y| = 2x + 2ln|x − 1| − 4

ln|y|  -  2ln|x − 1| = 2x − 4 

using property of loagarithms

ln|y|  -  ln|x − 1|² =2x − 4 

ln [|y| / |x − 1|² ]  =2x − 4 

y = (x − 1)² e^(2x − 4)

see this video for more explanation 

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