Use an integrating factor to solve

[Calu]

Homework Statement

Use an integrating factor to determine the general solutions of the following differential
equation:

dx/dt - 2/t = 2t³ + (4t²)(e^4t)

Homework Equations

R(x) = e^∫P(x).dx

The Attempt at a Solution

Usually the equation is in the form dx/dt + P(x)t = Q(x) but here I'm not sure what to do to find P(x) here as I have 1/t, t³ and t².

I'm also not sure how to go about finding a solution either. I know that once I have found the integrating factor, I have to multiply the original equation by R(x). After that I'm not sure what to do.

 

[jackarms]

Well, it looks like this equation is actually separable. Just move the -2/t over, and the right side will be entirely a function of t.

 

[Calu]

Well, it looks like this equation is actually separable. Just move the -2/t over, and the right side will be entirely a function of t.

I realized that whilst I was doing it, but the question specifically asks me to use an integrating factor. thanks for the reply though.

 

[LCKurtz]

Homework Statement

Use an integrating factor to determine the general solutions of the following differential
equation:

dx/dt - 2/t = 2t³ + (4t²)(e^4t)

Homework Equations

R(x) = e^∫P(x).dx

The Attempt at a Solution

Usually the equation is in the form dx/dt + P(x)t = Q(x) but here I'm not sure what to do to find P(x) here as I have 1/t, t³ and t².

I'm also not sure how to go about finding a solution either. I know that once I have found the integrating factor, I have to multiply the original equation by R(x). After that I'm not sure what to do.

Here, your independent variable is $t$. So your integrating factor is $R=e^{\int\frac 2 t~dt}$. Evaluate that and multiply through by it.

 

[Calu]

Here, your independent variable is $t$. So your integrating factor is $R=e^{\int\frac 2 t~dt}$. Evaluate that and multiply through by it.

Oh, I see. Thanks, I was being a bit stupid there.