Question

Given the following equations

y´(x)+y(x)=e^-x; 2y´(x)+y(x)=e^-x ; y´(x)+2y(x)=e^-x

^{a. Use the integration factor and find your solutions}

<sup>b. Discuss the trend behavior of the solutions and (x) as follows:
1. compare the graphs of y (x) with the graphs of F (x)
2. Compare the graphs of y (x) with respect to the coefficients of the differential equations</sup>

Answer 1

we know Huat +Prxy =G0) hai He general selutien SPENods yeprnidor Kper's yooTy0-x @yeJdx e-2da) -X ya= e araph ot ya fony >

hvaph ot Fixe e-2 Bo H you and Frx tendds to 22YO as T ends to ycx) cin d A-00 F(R y'ex)ya) e J y=e-Sida -2 ap 0.

ya- e da e +t

e esca) ya=e FX)= Bo th yot) and F) tendh to o つ( tends to and つ(→ ~Dマ FC1)0 cus

yx)tay() e - Pt)2, () =e- a G=e Jada -220 とと e Sendau -2つ( e araph ot Gvaph of F(x) = e- (on) Fla)e-

boh Here tends to aus a nd Fa) and tends to as 2 -D, Both behaues alike cuYue