Answer to Question #196776 in Calculus for desmond

Question

 1.By writing 3 cos x + 4 sin x = λ d dx (4 cos x + 5 sin x) + µ(4 cos x + 5 sin x), where λ and µ are constants

Answer 1

Given that,

$3cosx+4sinx=\lambda \dfrac{d}{dx}(4cosx+5sinx)+\mu(4cosx+5sinx)\\\Rightarrow 3cosx+4sinx=\lambda (-4sinx+5cosx)+\mu(4cosx+5sinx)\\\Rightarrow 3cosx+4sinx=(5\lambda+4\mu)cosx+(-4\lambda+5\mu)sinx$

Equating coefficient of sinx and cosx on both sides, we get

$5\lambda+4\mu = 3\ \ .....(i)\\-4\lambda+5\mu=4\ .....(ii)$

On solving equation (i) and (ii)

we get,

$\boxed{\mu=\dfrac{32}{41}}\ \ and\ \ \boxed{\lambda=-\dfrac{1}{41}}$