Q. 37 K5.0( 1 Vote )

Solve each of the following initial value problems:

, y = 0 when

Answer :

xi. , y = 0 when


Given and



This is a first order linear differential equation of the form



Here, P = 2 tan x and Q = sin x


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





We have



[ m log a = log am]



I.F = sec2x [ elog x = x]


Hence, the solution of the differential equation is,







Recall


ysec2x = sec x + c




y = cos x + c cos2x


However, when, we have y = 0






c = –2


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


y = cos x + (–2)cos2x


y = cos x – 2cos2x


Thus, the solution of the given initial value problem is y = cos x – 2cos2x


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