Question

Q3. [25 pt.] Find solution to the first order linear equation (ysinx – cos x esinx)dx + (cos x)dy = 0 under the initial value of y(0) = 3.

Answer 1

Ane) (Yenx cos? y ento) dx + (os ydy=0 divide eq by cosn. cos²x Sanx) + dy Y tanx -o dy tytana dy ce dx this first onden Linear Diffinential eq. general soln of +Plasy = Colul is dy da P(x) = tany cosx e sinx CO(x) = Spixsd Y.C estemde sores especonde gut Comet S cosx Snx Stomede de - y e log sex da tc Insect e Y(a) e Siosx can't Siose su secx da to Y(2) secx cose sind dx + c 4(x) Sect Let Sing-t cosa dx=dt You secx = set at + const you secx = et + Const y (2) secx = e + Const Sinx

2 y(x) = cosx e Sinx & C. cost 410)=3 Sinoo } = (osoo e + C. Cosoo 3 = It C (= 2 Sina + 2 cosa 4(x) = cosa e Ans
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