Q. 195.0( 1 Vote )

# Solve the following differential equations : (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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