First Order Differential Equations: Separable Variables

[ljharris0622]

I really need help with the following equations and solving them by using separable variables!!!

1) dy/dx + 4xy^2=0

2) dN/dt + N = Nte^t+2

3) dy/dt = x(1+y^2)

I really just need a step by step explanation for these because it's getting confusing. Please help me!

 

[ljharris0622]

For these particular problems, I'm mainly having trouble with separating the variables like for the following:

1) I simplified it like this

dy/dx = 4xy^2,
dy/y^2 = 4x dx

Then I got stuck from there...

2) I really did not know how to get started with this one.

3) I simplified it like this

dy/dx = x(1+y^2)
dy/1+y^2 = x dx

Then I also got stuck from there...

 

[galactus]

1) dy/dx + 4xy^2=0

\(\displaystyle \frac{dy}{dx}=-4xy^{2}\)

Separate variables:

\(\displaystyle \frac{dy}{y^{2}}=-4xdx\)

Now integrate:

\(\displaystyle \int \frac{1}{y^{2}}dy=-4\int xdx\)

2) dN/dt + N = Nte^t+2

\(\displaystyle \frac{dN}{dt}=Nte^{t+2}-N\)

Factor out N on the right:

\(\displaystyle \frac{dN}{dt}=N(te^{t+2}-1)\)

Now, separate variables:

\(\displaystyle \frac{dN}{N}=(te^{t+2}-1)dt\)

Now, integrate as in the previous problem.

3) dy/dt = x(1+y^2)

Are you sure there is no typo?.

Perhaps it is \(\displaystyle \frac{dy}{dx}=x(1+y^{2})\)

\(\displaystyle \frac{dy}{1+y^{2}}=xdx\)

Now, integrate. Remember, the left side integral involves \(\displaystyle tan^{-1}(y)\)

Can you do the integrations?.

 

[ljharris0622]

Yes I can do the integrations....thanks!