How can I solve the following exercise How can I solve the following exercise 
$$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$
$$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$

Let $u_1(s)$ be the Laplace transform of $\phi_1(x)$, and let $u_2(x)$ be the Laplace transform of $\phi_2(x)$. Since 
$$\varphi_1 = e^x - 1 * \varphi_1 + 4e^x * \varphi_2$$
and
$$\varphi_2 = 1 - e^{-x} * \varphi_1 + 1 * \varphi_2,$$
(where $*$ denotes the convolution product) taking the Laplace transform results in
$$u_1 = \frac{1}{s-1} - \frac{1}{s}u_1 + \frac{4}{s-1}u_2$$
and
$$u_2 = \frac{1}{s} - \frac{1}{s+1}u_1 + \frac{1}{s}u_2,$$
using the fact the Laplace transform of a convolution of two functions is the product of the Laplace transforms. This, in turn, yields a linear system of equations
\begin{align}
\left(1 + \frac{1}{s}\right)u_1 - \frac{4}{s-1}u_2 &= \frac{1}{s-1}\\
\frac{1}{s+1}u_1 + \left(1 - \frac{1}{s}\right)u_2 &= \frac{1}{s}\\
\end{align}
Solving for $u_1, u_2$, you'll obtain rational expressions for $u_1$ and $u_2$. After that, decompose the rational expressions by method of partial fractions, then compute the inverse Laplace transform of each partial fraction to obtain $\varphi_1$ and $\varphi_2$.
