Question

4. (10 points) Solve the given IVP: y'"' + 8y" +22y' + 20y = 0; y(0) = 0, y'(0) = 1, y" (0) = 2.

Answer 1

d² y + 8 da The given differential equation is y" + 8y" +224 +20y = 0 dy +22 dy dx² dx² 20y=0 y= A eme be a trial solution. Therefore, auxiliary equation is, m² +8m² +22m + 20 m² + 2m + 6m² + 12 m+ 10m+2020 > m (m + 2) + 6 m (m + 2) + 10 olm+2) -0. (m² + 6m + 10) (m + 2) = 0. => m + 2) = 0 m² tomt 10=0. ΟΥ m² + 6 m+10=0. - 6 16-6)? 4 x 1 x 10, Then, 6 + ✓ 36-40 ma 2 ma 2 = m = 6 + 2i 2 = m = -3 + i.

Hence, m² + 6 m+10=0 m=-3ti or -3-i m+2=0 m=-2. Jos m =- 2 - 3 ti, -3-1 Hence, cons which are Thus, general solution of his y = C, e-24 +[C, Cosa + Co Bina] where c,,C2, C3 are tants to be determined by the given initial conditions. y = C, e- 2x + (-3 [C2 Cosa + Ca sina] Differentiating with respect to x, we obtain, dy = 62c.) e 2x +(-3) e-3x [C2 Cosa + Ca sinn] + e- 3* [- C2 sinn da & C3 Cosa 1

dy = (-20) e- dx ( -1) e-2x + (-3x [£3 C2 + (3) Cosa + (-3C3 - (2) sin a #1. Again, differentiating with respect to x, we obtain, dy [[3 C2 + C₂). €2C,).(-2) e- 2x + (-3). e-3n dxz cosx + (-3 Cg - Cz) sinn] + e-3x [(-3 C2 + Cg) (- sinx) + (-3C3 - C₂). Cosa dy (AC) e-2x + e [ 2 Z3 ( 9 C2 - 3 C3 -3 C3 -C2), cosx + ( 9 C3 + 3 C2 +3 € ₂ 0 ) ] e-3a dxz sina

daz -32 dý = (40) e-2x + [(8C2 -663) Cosx + (8 C₂ +6C2) sina It is that, given y(0) 0 → 4,2-2.0.0-30 [catos O + C, Bin 0 ]-- C, + eo. [C₂ + 0 = 0. - Cat + C2 = 0 y (0) - 1 dyl da → (201)ę 2.0 + e 3.0 [€ 3 C ₂ + C₂). Los o +(-3C₂-C₂). sino 0]-1 - 2 C, + (-36₂ + (3) = 1 -20, -- 3 C2 + 3 = 1 3

y" (O) = 2 D. 2. + e-3.0 - (46) e 2.0 [(862 - 6 C3). Los o + ( 8 C3 + 6. C2). sin o] = ? = 2. - 4 C, + 8 C2 -663 Hence, 3 C, & C2 +0. C3 = -20, -3C₂ + 1. C₂ 1 4 C, + 8C2 = 2 6 C3 4 42 -2 - 3 1 4 8 18- 8 ] -1. [12-4 - 10 - 8 = 2. A 1) - 2

A2 = 1. (-6-2) - - 2 4 2 - 8 43 2 - 3 8 2 1. [-6-8] - 1. [-4-4 - 14 + 8 = -6 Hence, by Cramer's rule, Ai Ci 8 2 24 A 42 8 C2= P. -4 43 Cze 2 – 3 4 2 Hence, general solution is y = 4 e 22+ er [~ 4 Cosx + (-3) sin a]