Linear first order differential equations - just a question

[DivGradCurl]

I thought the general solution of the linear first order differential equation

$$y ^{\prime} + p(t)y = g(t), \qquad y\left( t_0 \right) = y_0$$

were

$$y = \frac{1}{\mu (t)} \left[ \int \mu (s) g(s) \: ds + \mathrm{C} \right],$$

where

$$\mu (t) = \exp \int p(s) \: ds$$

However, I have found this

$$y = \frac{1}{\mu (t)} \left[ \int _{t_0} ^t \mu (s) g(s) \: ds + y_0 \right],$$

where

$$\mu (t) = \exp \int _{t_0} ^t p(s) \: ds$$

Can anybody please clarify that? I don't see why those integral limits should be there. Any help is highly appreciated.

 

[geosonel]

general solution correct.

however, in this case, integration limits needed for initial condition:

$$y\left( t_0 \right) = y_0$$

that is:

$$\mu (t_0) = \exp \int _{t_0} ^{t_0} p(s) \: ds = \exp(0) = 1$$

$$y(t_0) = \frac{1}{\mu (t_0)} \left[ \int _{t_0} ^{t_0} \mu (s) g(s) \: ds + y_0 \right] = \frac {1} {1} \left [(0) \ + \ y_0 \right ] \ = \ y_0$$

nothing really complicated here

 

[thepatientmental]

The general solution of a linear first order differential equation is given by the formula y = \frac{1}{\mu (t)} \left[ \int \mu (s) g(s) \: ds + \mathrm{C} \right], where \mu (t) = \exp \int p(s) \: ds. This formula is correct and can be used to find the general solution to any linear first order differential equation. However, in the specific case where we have an initial condition y(t_0) = y_0, the value of \mathrm{C} can be determined using this condition. This results in the second formula y = \frac{1}{\mu (t)} \left[ \int _{t_0} ^t \mu (s) g(s) \: ds + y_0 \right], where \mu (t) = \exp \int _{t_0} ^t p(s) \: ds. The integral limits are necessary in this case because we are integrating from the initial time t_0 to the current time t. The presence of the initial condition y(t_0) = y_0 also affects the value of \mathrm{C}, hence the difference in the formula. Both formulas are correct and can be used depending on the given initial condition. I hope this clarifies the confusion.