1st Order Diff. Equations - Need Help

[travh2007]

Directions: Rewrite the following equation as a linear 1st Order Differential Equation in standard form. (Standard form = dy/dx + P(x)y = f(x))

dy/dx = xy(1+y³)

My problem is that there are too many y variables in this equation, and the standard form only has room for 1. So I figured I could write this in terms of x, since I only have one of them.


So..I tried to rewrite it by changing the (dy/dx) to (dx/dy) and then putting it in standard form.

So basically I changed the original equation to this:

dx/dy = xy(1+y³)

and then rearranged the equation to look like this:

dx/dy - (y+y³)x = 0

Is this the correct way of going about this problem?

 

[Mark44]

Directions: Rewrite the following equation as a linear 1st Order Differential Equation in standard form. (Standard form = dy/dx + P(x)y = f(x))

dy/dx = xy(1+y³)

My problem is that there are too many y variables in this equation, and the standard form only has room for 1. So I figured I could write this in terms of x, since I only have one of them.


So..I tried to rewrite it by changing the (dy/dx) to (dx/dy) and then putting it in standard form.

So basically I changed the original equation to this:

dx/dy = xy(1+y³)

The step above isn't valid.

If dy/dx = xy(1 + y³), then dx/dy = 1/[ xy(1 + y³)]

and then rearranged the equation to look like this:

dx/dy - (y+y³)x = 0

Is this the correct way of going about this problem?

I don't see any way of writing dy/dx = xy(1 + y³) as dy/dx + P(x)y = f(x).

If you know the technique of separation of variables, the DE can be solved that way.

 

[Saladsamurai]

What Mark said. But also: my first instinct is re read the instructions: "rewrite as a *linear* ode in standard form." Your current equation is certainly not linear in y; however, I believe it is Bernoulli, which means it can be converted to a linear ODE. But I am on the train; so I could be mistaken.