$2$

Formula $1$ point

Consider a non$-$homogeneous $1$st order linear ODE:

$x{y}^{\text{'}}+y=2\mathrm{.}5e^x$

The general solution has structure $y=C{y}_c+y_p$ where

$y_c($Complementary solution$)$ is a solution to the associated homogeneous Equation: $x{y}^{\text{'}}+y=0$ with an arbitrary constant $C.$

$y_p$ is ANY particular solution to $x{y}^{\text{'}}+y=={?}^{}{?}^{\text{'}}{e}^x.$

For $x_1=1\mathrm{.}3,$ find $\frac{y_p(x_1)}{y_c(x_1)}$

Include at least $3$ decimal places in each step in your calculation.

Use $=3\mathrm{.}14159,$ and $e=2\mathrm{.}71828$