Consider the initial value problemy″−20y′+100y=e−10x1+x2,y(0)=−5,y′(0)=−8. (1) Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential...

Consider the initial value problemy″−20y′+100y=e−10x1+x2,y(0)=−5,y′(0)=−8.

(1) Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y″−20y′+100y=0 is the function yh(x)=C1 y1(x)+C2 y2(x)=C1  +C2.
Note: The order in which you enter the answers is important; that is, C1f(x)+C2g(x)≠C1g(x)+C2f(x).

(2) The particular solution yp(x) to the differential equation y″+20y′+100y=e−10x1+x2 is of the form yp(x)=y1(x) u1(x)+y2(x) u2(x) where u1′(x)=  and u2′(x)= .

(3) The most general solution to the non-homogeneous differential equation y″−20y′+100y=e−10x1+x2 is

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Extracted text: Consider the initial value problem -10x e y' – 20y/ + 100y = у (0) — —5, у/ (0) — —8. 1+ x² (1) Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" – 20y + 100y = 0 is the function Ул (а) — Сі у(х) + Сә у2(2) — С e^10х) +C2 хе^(10х) = Note: The order in which you enter the answers is important; that is, Cif(x) + C29(x) # C19g(x) + C2f(x). -10z e (2) The particular solution yp(x) to the differential equation y/" + 20y + 100y = is of the form yp(x) = y1(x) u1(x) + y2(x) u2(x) where u (x) 1+ x? (-xe^(-20x))/(1+x^2) and u (x) (e^(-20x))/(1+x^2) -10x e (3) The most general solution to the non-homogeneous differential equation y" – 20y + 100y= is 1+ x? y = 1/(e^(20x)(1+x^2)) + e^(10x) 1/(e^(20x)(1+x^2)) dt+ xe^(10x) -X/(e^(20x)(1+x^2)) dt