General Solution

[Sjsusu]

Hello, my calculus teacher recently assigned us these problems but he has not yet taught Differential Equations. I have been able to solve most of the other questions but I can't seem to figure this one out.
I have cross multiplied and integrated to find (1/4)sin(4y)-(1/8)sin(8x)=c, which matches none of the answer choices given. Am I missing something?
Thank you for taking the time to read this post, any help is much appreciated.

 

[topsquark]

Again use
[math]\dfrac{dy}{dx} = \dfrac{cos(8x)}{cos(4y)}[/math]
[math]cos(4y) ~ dy = cos(8x) ~ dx[/math]
[math]\int cos(4y) ~ dy = \int cos(8x) ~ dx[/math]
See where you can go from here.

-Dan

 

[Sjsusu]

Hello, after I integrated, I got:
(1/4)sin(4y)=(1/8)sin(8x)+c
Is this right?

 

[topsquark]

Hello, after I integrated, I got:
(1/4)sin(4y)=(1/8)sin(8x)+c
Is this right?

Looks good!

-Dan

 

[Sjsusu]

Hello, after I integrated, I got:
(1/4)sin(4y)=(1/8)sin(8x)+c
Is this right?

Thanks the only issue I have is when I isolate c front he equation, my answer doesn't match any of the choices. However c is proportional to (1/4)sin(4y)-(1/8)sin(8x)=c, should I choose c?

 

[topsquark]

Thanks the only issue I have is when I isolate c front he equation, my answer doesn't match any of the choices. However c is proportional to (1/4)sin(4y)-(1/8)sin(8x)=c, should I choose c?

[math]8 \cdot \left ( \dfrac{1}{4} ~ sin(4y) \right ) = 8 \cdot \left ( \dfrac{1}{8} ~ sin(8x) + c \right )[/math]
[math]2 ~ sin(4y) = sin(8x) + 8c[/math]
Since c is an arbitrary constant, so is 8c. We can just call this C.

So
[math]2 ~ sin(4y) - sin(8x) = C[/math], which is choice c).

-Dan

 

[Sjsusu]

[math]8 \cdot \left ( \dfrac{1}{4} ~ sin(4y) \right ) = 8 \cdot \left ( \dfrac{1}{8} ~ sin(8x) + c \right )[/math]
[math]2 ~ sin(4y) = sin(8x) + 8c[/math]
Since c is an arbitrary constant, so is 8c. We can just call this C.

So
[math]2 ~ sin(4y) - sin(8x) = C[/math], which is choice c).

-Dan

Thank you!

 

[HallsofIvy]

If this is a Calculus course and your teacher "has not yet taught Differential Equations" perhaps you are not expected to actually solve the equations! Instead differentiate each of the suggested solutions, put the into the equation and see which satisfies the equation.

For example, (A) is sin(4y)- sin(8x)= C.
Differentiating, 4 cos(4y)(dy/dx)- 8 cos(8x)= 0
4 cos(4y)(dy/dx)= 8 cos(8x)
dy/dx= 2 cos(8x)/cos(4y)
No, that's not quite correct so try the next one.