Question

1. Find the general solution of the equation:

  y'' + 5y = 0

2. Find the general solution of the equation and use Euler’s formula to place the solution in terms of trigonometric functions:

y'''+y''-2y=0

3. Find the particular solution of the equation:

y''+6y'+9y=0

where

y1=3

y'1=-2

Part 2: Nonhomogeneous Equations

5. Find the general solution of the equation using the method of undetermined coefficients:

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6. Now find the general solution of the equation using the method of variation of parameters without using the formula from the book:

$pc1OkSxeIAAAAASUVORK5CYII=$

7. Now find the general solution of the equation using the method of variation of parameters using the formula in the book:

y''-y'=5sin⁡(2x)

Answer 1

y"+syo The auxiliary eqeen in m²+5=0 marsi Therefore the general soln in a y (2) = q con(5x) + C sim (5x) where e, and C2 are conforto. +2 m-220 Again, y'"+y" - 2y =0 The auxiliary equation in : m² + m²-2 20 m3-m²2m² -2m sty m m -1) +2004 -) + 2m-d) 20 . (m2+2 m +2)(m-1).0 i masao or m²+2m+220 mai m -2.754-8 2 - -2 fli 2 Therefore, the general soll is. y(a) Gje+ ē" (e corx+ eg sina) whene 1, C2, C₂ ane compact

Atow , y4+ 6y'+9420 in - the auxiliary sequen m² + 6 m+q 50 » (m + 3)2=0 osm=-9,-3 Therefore, the general solution in : y (a) = (es + ex) e 32 Geinen, y (2) = 3 and y' (2)= -2 .: (+)73 = e 3 ---D Also, y' (2) = -3 9 0 32 2 e 32 -3.,we 32 = (-34 +62) 32 36,2832 . y' (t) = (+34 + 2)23 -- 36, e3 * -2 = -3403 - 2e2e3 ♡ x3 + ② x 1 giver 39€ 3+3C, 639 -3% 83 - 26ē3.-2 i. = 7e3 iga 3 e3 – 7 3 2-4e3 .*y(a) z (-4+4x)32+3