Need help with integration using integrating factor

[Millie]

Hi all

Need some help with integration steps.

I am trying to find the integration result of d/dt P(t) +(σ+ρ)P(t) = σ
This is then multiplied by the integrating factor e^((σ+ρ)t) to give the next line:
e^((σ+ρ)t) d/dt P(t) + (σ+ρ) e^((σ+ρ)t) P(t)= σ e^((σ+ρ)t)
We then integrate both sides with respect to t
The RHS gives e^((σ+ρ)t) P(t)
I dont understand how this is derived. How did we get from e^((σ+ρ)t) d/dt P(t) + (σ+ρ) e^((σ+ρ)t) P(t) to e^((σ+ρ)t) P(t).
I would be grateful if I could be taken through the breakdown of the steps.

Thank you kindly.

 

[Dave Johnson]

Hi Millie

The integrating factor method is a reverse engineering of the product rule, so what we have in your example is:

$$ u = e^{(\sigma + \rho) t} $$
$$ v = P(t) $$

Which, by applying the product rule, we can write as:

$$ u dv/dt + v du/dt = d(uv)/dt $$

Then integrating both sides with respect to t yields uv.

Hope this helps,

Dave

 

[Millie]

Hi Millie

The integrating factor method is a reverse engineering of the product rule, so what we have in your example is:

$$ u = e^{(\sigma + \rho) t} $$
$$ v = P(t) $$

Which, by applying the product rule, we can write as:

$$ u dv/dt + v du/dt = d(uv)/dt $$

Then integrating both sides with respect to t yields uv.

Hope this helps,

Dave

Yes, thanks very much! Much appreciated