Q. 2 B3.6( 8 Votes )

# For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

y = e^{x} (a cos x + b sin x) :

Answer :

It is given that y = e^{x}(acosx + bsinx) = ae^{x}cosx + be^{x}sinx

Now, differentiating both sides w.r.t. x, we get,

Now, again differentiating both sides w.r.t. x, we get,

Now, Substituting the values of ’ and in the given differential equations, we get,

LHS =

=2e^{x}(bcosx – asinx) -2e^{x}[(a + b)cosx + (b –a ) sinx] + 2e^{x}(acosx + bsinx)

=e^{x}[(2bcosx – 2asinx) - (2acosx + 2bcosx) - (2bsinx – 2asinx) + (2acosx + 2bsinx)]

= e^{x}[(2b – 2a – 2b + 2a)cosx] + e^{x}[(-2a – 2b + 2a + 2bsinx]

= 0 = RHS.

Therefore, the given function is the solution of the corresponding differential equation.

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