- #1

Lambda96

- 189

- 65

- Homework Statement
- Show that the solution of a linear differential equation with boundary condition ##f(0)=f_0## has the following form ##f(x)=f_0 exp\biggl( \int_{0}^{x}ds ~g(s) \biggr)+\int_{0}^{x}ds ~h(s)exp\biggl( \int_{s}^{x}dr ~g(r)\biggr)##

- Relevant Equations
- none

Hi,

unfortunately I have problems with the task d and e, the complete task is as follows:

I tried to form the derivative of the equation ##f(x)##, but unfortunately I have problems with the second part, which is why I only got the following.

$$\frac{d f(x)}{dx}=f_0 g(x) \ exp\biggl( \int_{0}^{x}ds \ g(s) \biggr)+?$$

I then tried another approach and first wanted to get rid of the integrals in the exp terms

$$f_0 exp\biggl( G(x)-G(0) \biggr)+\int_{0}^{x}ds \ h(s) exp \biggl( G(x)-G(s) \biggr) f(x)dx$$

$$f_0 e^{G(x)-G(0)}+e^{G(x)}\int_{0}^{x}ds \ h(s) e^{-G(s)}$$

Unfortunately, I now have problems again with the derivative for the second term using this method.

unfortunately I have problems with the task d and e, the complete task is as follows:

I tried to form the derivative of the equation ##f(x)##, but unfortunately I have problems with the second part, which is why I only got the following.

$$\frac{d f(x)}{dx}=f_0 g(x) \ exp\biggl( \int_{0}^{x}ds \ g(s) \biggr)+?$$

I then tried another approach and first wanted to get rid of the integrals in the exp terms

$$f_0 exp\biggl( G(x)-G(0) \biggr)+\int_{0}^{x}ds \ h(s) exp \biggl( G(x)-G(s) \biggr) f(x)dx$$

$$f_0 e^{G(x)-G(0)}+e^{G(x)}\int_{0}^{x}ds \ h(s) e^{-G(s)}$$

Unfortunately, I now have problems again with the derivative for the second term using this method.