# Differential equation w/ cos and sin

• MHB
• Emjay
In summary, the conversation is about a question involving an unknown function y(x), where x^2 cos y + sin(3x-4y) =3. The question is whether this is a differential equation or a purely functional equation. Suggestions were made to use trig identities and implicit differentiation to solve for $\dfrac{dy}{dx}$.
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 :)

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 $y$.

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.

Emjay said:
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.

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes the relationship between a function and its rate of change.

## 2. What is the role of cosine and sine in a differential equation?

Cosine and sine are trigonometric functions that can be used to model periodic behavior in a differential equation. They are often used to describe the movement of a system over time.

## 3. How are differential equations with cosine and sine solved?

Differential equations with cosine and sine can be solved using various methods, such as separation of variables, substitution, and power series. The appropriate method depends on the specific equation and its initial conditions.

## 4. What are some real-world applications of differential equations with cosine and sine?

Differential equations with cosine and sine have many applications in physics, engineering, and other fields. They can be used to model oscillating systems, such as a pendulum or a spring, as well as phenomena such as sound waves and electrical circuits.

## 5. Are there any limitations or challenges in using differential equations with cosine and sine?

One limitation of using differential equations with cosine and sine is that they can only model linear systems, meaning the relationship between the function and its derivatives is linear. They may also be challenging to solve analytically for more complex systems, requiring numerical methods instead.

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