Solve each of the following initial value problems:
, y(1) = 0
, y(1) = 0
Given and y(1) = 0
This is a first order linear differential equation of the form
Here, and
The integrating factor (I.F) of this differential equation is,
We have
[∵ m log a = log am]
∴ I.F = x–1 [∵ elog x = x]
Hence, the solution of the differential equation is,
Recall
∴ y = –log x – 1 + cx
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