How can I solve this integral equation by converting it to a differential equation Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$
How can I solve this integral equation by converting it to a differential equation 

Given $\displaystyle y(x)=e^{-x}\cos(x)-\cos(x) \cdot \int_{0}^{x}e^{-x+t}y(t)dt...........(1)$
Using Differentitation Under Integral Sign.
Given $\displaystyle \frac{d}{dx}\left(y(x)\right) = \frac{d}{dx}\left[e^{-x}\cdot \cos x\right] - \frac{d}{dx}\left[\cos x \cdot \int_{0}^{x}e^{-x+t} \cdot y(t)dt\right]$
We Get $\displaystyle y'(x) = -e^{-x}\cdot \sin x+\cos x\cdot e^{-x}-\cos x\cdot y(x)+\int_{0}^{x}e^{-x+t}\cdot y(t)dt\cdot \sin x $
Now Put Value of $\int_{0}^{x}e^{-x+t}\cdot y(t)dt$ from $\bf{1^{st}}$ equation.
$\displaystyle y'(x) = -e^{-x}\cdot \sin x+\cos x\cdot e^{-x}-\cos x\cdot  y(x)+\left[\frac{e^{-x}\cdot \cos x-y(x)}{\cos x}\right]\cdot \sin x$
You get an Differential equation
