# First order Ordinary Differential Equation

1. Sep 14, 2009

### PAR

1. The problem statement, all variables and given/known data
Solve the differential equation:

$$y' = 8sin(4yt) ; y(1) = 4$$

2. Relevant equations

3. The attempt at a solution

The integrating doesn't apply because I can't get the equation into:
y' + p*y = f(x) form

Also, I have tried separating variables, but I can't get the y inside of the sin(4yt) outside of the sin(). Basically, I need some help to get started.

Last edited: Sep 15, 2009
2. Sep 15, 2009

### Avodyne

Try changing variables: y(t)=z(t)/t.

3. Sep 15, 2009

### PAR

Correction! The differential equation should be y' = 8sin(4yt) instead of y'=8y$$^{3}$$sin(4yt)

I've tried substitution where y(t) = z(t)/t

This gave me:

$$y' = z'/t - z/t^{2} = 8sin(4z)$$

The problem now is that I still have a function of z on the right hand side of the equation, and I don't know how to combine it with the left hand side to give me an a nice integrating factor or separation of variables. I was thinking I could divide the whole thing by sin(4z) anyway, and then integrate with respect to t. The z' would be integrable and so would the 8, but the resulting -z/(t$$^{2}$$sin(4z)) I don't know how to integrate with respect to t. Great help so far, but could still use more :).

4. Sep 15, 2009

### Gregg

$$\frac{dy}{dt} = 8\sin (4yt)$$

$$y=\frac{z}{t} \Rightarrow 8 \sin (4yt) = 8 \sin (4z) = \frac{dy}{dt}$$

$$y=\frac{z}{t}$$

$$\frac{dy}{dt}=-\frac{dz}{t^2 dt}$$

$$\frac{dz}{dt} = -8t^2\sin (4z)$$

$$\int \frac{dz}{\sin (4z)} = \int -8t^2 dt$$

Maybe that will work?

5. Sep 15, 2009

### PAR

Don't you have to apply the product rule when you differentiate y?

Since y = z/t = zt$$^{-1}$$ and z is a function of t by definition shouldn't

y' = z'/t - z/t$$^{-2}$$ instead of y' = $$-\frac{dz}{t^2 dt}$$

6. Sep 15, 2009

### Gregg

Oh yeah god, use the product rule :\$.

7. Sep 15, 2009

### PAR

Thank you Gregg and Avodyne for the help, but I don't have a solution to this problem. Is there a different way to solve first order ODE other than separation of variables and the integration factor? Please, I need some help.