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mcastillo356

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- TL;DR Summary
- I can't understand why the result of this integral equation is the differential equation of exponential growth

Hi, PF

There goes the solved example, the doubt, and the attempt:

PD: Post without preview

Greetings!

There goes the solved example, the doubt, and the attempt:

**Example 8**Solve the**integral equation**##f(x)=2+\displaystyle\int_4^x\,f(t)dt##.**Differentiate the integral equation ##f'(x)=3f(x)##, the DE for exponential growth, having solution ##f(x)=Ce^{3x}##. Now put ##x=4## into the integral equation to get ##f(4)=2##. Hence ##2=Ce^{12}##. So ##C=2e^{-12}##. Therefore, the integral equation has solution ##2e^{3x-12}##.***Solution***Doubt**Why is the DE for exponential growth when we differentiate the integral equation to get ##f'(x)=3f(x)##?**Attempt**It is a shot in the dark. Many natural processes involve quantities that increase or decrease at a rate proportional to their size. All of these phenomena, can be modelled ##\displaystyle\frac{dy}{dt}=ky##, this is, the**differential equation of exponential growth or decay**.PD: Post without preview

Greetings!