MHB Differential equation w/ cos and sin

[Emjay]

It would be wonderful if someone could please help with the following question as I don't even know where to begin

y=y(x), where x^2 cos y + sin(3x-4y) =3Thank you :)

 

[S.G. Janssens]

Since you wrote "differential equation", I would like to ask whether you are sure this is indeed what you want to solve?

Namely, I would call this a purely functional equation, i.e. an equation involving an unknown function $y$ of $x$, but no derivatives of $y$.

 

[HallsofIvy]

As Krylov said, this is NOT a "differential equation". I would start by using trig identities to get every thing in terms of sine and cosine of y only together with sine and cosine of x.

 

[skeeter]

It would be wonderful if someone could please help with the following question as I don't even know where to begin

y=y(x), where x^2 cos y + sin(3x-4y) =3

... maybe you're looking to determine $\dfrac{dy}{dx}$ using implicit differentiation?

$\dfrac{d}{dx} \bigg[x^2 \cos{y} + \sin(3x-4y) =3 \bigg]$

product rule & chain rule ...

$-x^2\sin{y} \cdot \dfrac{dy}{dx} + 2x\cos{y} + \cos(3x-4y) \cdot \left(3 - 4\dfrac{dy}{dx}\right) = 0$

If my assumption is correct, then complete the algebra necessary to isolate $\dfrac{dy}{dx}$, if not ... oh well.