A homogeneous second$-$order linear differential equation, two functions $y_1$ and $y_2,$ and a pair of initial conditions are given. First verify that $y_1$ and $y_2$ are solutions of the differential equation. Then find a particular solution of the form $y=c_1{y}_1+c_2{y}_2$ that satisfies the given initial conditions. Primes denote derivatives with respect to $x.$

$y^{\text{'}\text{'}}-2y^{\text{'}}+2y=0$;$y_1=e^{x}cosx,y_2=e^{x}sinx$;$y(0)=19,y^{\text{'}}(0)=5$

Why is the function $y_1=e^{x}cosx$ a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice.

A$.$ The function $y_1=e^{x}cosx$ is a solution because when the function and its indefinite integral, $$ are substituted into the equation, the result is a true statement.

B$.$ The function $y_1=e^{x}cosx$ is a solution because when the function, its first derivative, $y_1^{\text{'}}=$ and its second derivative, $y_1^{\text{'}\text{'}}=,$ are substituted into the equation, the result is a true statement.