Problem $2$

$(8$ Grades$)$:

Translate the following sentences into a logical expression

$1-$ You get an A if you score at least $50\%$ on the midterm or you submit a HW

$2-$ Scoring at least $50\%$ on the midterm or submitting a HW is necessary for getting

an A

$3-$ You get an A if and only if you score at least $50\%$ on the midterm or you submit a

HW

$4-$ You get an A only if you score at least $50\%$ on the midterm or you submit a HW

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{y^2+4}{y+2}d{y}_{\text{d}\text{i}\text{v}\text{i}\text{s}\text{i}\text{o}\text{n}\text{l}\text{o}\text{n}\text{g}}$

$\frac{dy}{dx}=\frac{2y^4+x^4}{x{y}^3}$

Q$1$: Determine the form of a particular solution for

$y^{\text{'}\text{'}}+3y^{\text{'}}+2y=10e^{-2x}sin3x-2e^{-x}+5x^2+x^{3}cos2x-7$

$(5$ points$)$ Given that $y_1=x^4$ is a solution of the differential equation

$y^{\text{'}\text{'}}-\frac{7}{x}{y}^{\text{'}}+\frac{16}{x^2}y=0,x>0,$