Q. 195.0( 1 Vote )

Solve the following differential equations :


Answer :

(i) If a differential equation is ,


then y(I.F) = ∫Q.(I.F)dx + c, where I.F = e∫Pdx


(ii) ∫tanxdx = log|secx| + c



(iv) ∫cosxdx = sinx + c



given:-



This is a linear differential equation, comparing it with



P = tanx, Q = x2cos2x


I.F = e∫Pdx


= e∫tanxdx


= elog|secx|


= secx


Solution of the equation is given by


y(I.F) = ∫Q.(I.F)dx + c


ysecx = ∫(x2 cos2x(secx)dx + c


ysinx = ∫(x2 cosxdx + c


ysecx = x2∫ cosxdx–∫(2x cosxdx)dx + c


using integrating by parts


y(secx) = x2sinx–2∫x2 sinxdx + c


y(secx) = x2sinx–2(x∫ sinxdx–∫ sinxdx)dx + c


y(secx) = x2sinx + 2xcosx–2sinx + c


y = x2sinxcosx–2xcos2x–2sinxcos2x–2sinxcosx + ccosx


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