Q. 37 B5.0( 1 Vote )

Solve each of the following initial value problems:

, y(1) = 0

Answer :

, 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


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