Solving Differential Equation: ψ'(t)=β((l(t))/(w[L(t)]))ψ(t)

[Economist2008]

Hello,
I'm wondering if you could find a solution to the following differential equation


ψ'(t)=β((l(t))/(w[L(t)]))ψ(t)-β((l(t))/(w[L(t)]))

where L(t)=∫l(t)dt
β=((∂w)/(∂L(t)))
β is a constant

I've been tryint to find a solution for ages. Please help!

 

[tiny-tim]

Welcome to PF!

Hello,
I'm wondering if you could find a solution to the following differential equation


ψ'(t)=β((l(t))/(w[L(t)]))ψ(t)-β((l(t))/(w[L(t)]))

where L(t)=∫l(t)dt
β=((∂w)/(∂L(t)))
β is a constant

I've been tryint to find a solution for ages. Please help!

Hi Economist2008! Welcome to PF!

Your equation is very confusing.

Can you tell us what the context is?

If dw/dL(t) = β is constant, then w = βL.

 

[Economist2008]

Thanks for replying Tim,

Ok let me tell you the whole story of my problem. The variable I would like to solve for is ψ(t), which is a co-state variable for the state variable L(t). ψ(t) has the meaning of what an additional increment of labour supply is worth for the future, because wage goes up with L(t).

The context is that I want to find the optimal time to get children. l(t) is the instantaneous labour supply of some female at time t. L(t) is the cumulative amount of labor up to time t. The wage w(L(t)) depends positively on the cumulative labour supply. L(t) can be thought of being a measure for experience. Wage goes up the more working experience the female has.

If you know the answer, I would be very happy and could continue with the paper. Thanks. :-)