Question

2. (4 points each) Find the particular solutions for the following differential equations. (a) y" – 4y' = et (b) y' +9y = sec 3t tan 3t

This is for a Differential Equation Class

Answer 1

following differential Find the particular solution for the equations (a) YH- 4y = ezt For pasiticular solution 9,6+) - Robes D-4D ezt 11 ezt 4C2) Ipht = I ezt -4 (b) y + y = fec3t tangt For particular solution. (by Variation of Parater). Ypht) = 2, 104, (+) + 2 (H) 4₂(4) homogeneous solution of y"+gy=0 946) = G Sin3t + C₂ Cust Splt) = 4, Lt Sinat + Q₂ (4) Losat &, (t) = T 866).9, + .dt WLt swt) - Wmonskian T y, it) and y,ct X(t) = secat tant

T, (+) = Oct). 4,4 •dt W+) Cost Sint WG) = =-3 3 usat -3 sinat = پاه (seczt) (tanzt) wszt dt -3 11 - 3 l tanzt dt : s sinat de CoS3t let cost = u 3 Sinat dt = du N = -1/3 logu - I log I cost = 2,64) = log1sec 3+)

د (دره xit.y, t) dt Wit) (ec3t) (tan3 t) Sh3t dt 5 ) F - = X Sin3t Cost .xsinat de cost f = AT sin²3t dt cas²t 2 (t) = -7 - Stanist at = - 1 ſ (sec 3t -t) at = -7 / sechst at that - tog let st= 4 at du Sata sec²udu + = + t -- tanu 3 K244) = - tar -1 tanst t 404) = ( log/sec3+)) Sin3t +(tanat + fanit + t cost 3 5)