Solve each of the following initial value problems: x(dy/dx) - y = log x, y(1) = 0

Question

Solve each of the following initial value problems:

\(x\frac{dy}{dx}-y=log\,x,\,y(1)=0\)

Answer 1

This is a first order linear differential equation of the form

The integrating factor (I.F) of this differential equation is,

Hence, the solution of the differential equation is,

However, when x = 1, we have y = 0

⇒ 0 = –log 1 – 1 + c(1)

⇒ 0 = –0 – 1 + c

⇒ 0 = –1 + c

∴ c = 1

By substituting the value of c in the equation for y, we get

y = –log x – 1 + (1)x

⇒ y = –log x – 1 + x

∴ y = x – 1 – log x

Thus, the solution of the given initial value problem is y = x – 1 – log x