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) :

It is given that y = e^x(acosx + bsinx) = ae^xcosx + be^xsinx

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.