Question

dey The general solution of the differential equation 3 dy dx + 2y = ex dx2 Select one: y = Ae* - Be2x + xex a. y = Ae*+ Be2x -xe-x b. y = Ae* + Be2x - xex C. y = Ae X+ Be-2x + xex d.

can someone solve this differential equation. I'll definitely leave good remarks

Answer 1

dy : 3 dy da +2y = ez dx² 00 The auxiliary equation is given by m²_.3m+2=0 -) (M-1) (-2)=0 of the auniliany Thus m=d, 2 are the roots equation Hence the complementary solution is Yc () = A et Be x 2x li where A and B are arbitrary constants. Now we find the particular solution to (1), which is given by wher D is the yp (2) (ek) operator DE o D²_3D +2 operation da 1 = (e) CD-1) (D-2) 1 D-1) (0-2) yleeg ek an e (0-1) (1-2) The we know that 1 (8) f (D) fra) i if fra) to it en la an a (0-1) =.ee (D+1-1) 1 (1) "As we know that 1 ease y (x) = e V(x) FCD) scota)

Then Y p.(x) = -e 1 / 6 (1) ol -e".x represents integration with respect tox Thus the general solution of the given differential equation is giv iven by y cu) - Yc(x) + Yp (2) => Y (%) = A ² + B e 2x xee Thus option . (c) is correct.