How can I solve the following exercise

Question

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

Answer 1

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 $\varphi_2$.