Question

1. Find the solution to the IVP : yy - x = 1, y (0) = 2 2. Find the general solution to the exact DE: e* dx – ydy = 0 3. Use ji = cos y to find an EXPLICIT solution to: (tan y)dx + xdy = 0

Answer 1

1 solue I UP y lol=2 Уу" x = 1 solm 1 + 2 y de la (1+x) dx hp h integration Slo+x) dx +C/2 fp hf y% => (1+x)? + 등 2 2 + => (1+x)2 + c given 2 that y com from @ (1+0)2 + c than 2 = > 2 - 4- 3 С then from @ у? + 3 (1+x)? + x²+ 2 x + 2x + 4 => у

on taking ive sign ylx) Vx+ 2x+4 - 2 y lo = J4 Yol = 2 72 but 2 (given) yo) - is so Required Som 2 Y(x) x + 2x + 4 solve et da y dy O Som - ex da = ydy on integration ex y% + с which is required gestreal som

Н -1 п С. dre Y <= da 거 ҡР HP х 1 . 1 - к п Y (" ҮР ИР х- 1 - 1 (= 1 6) Кокт hros «Р t HP ҡ — қу- о к хо) — 1 -> Р [roo huis HP x — (= о simy «Р К- Т. hu Cust + Р Крх Swan t о- hurst C) huis Р Кр da HP xp HP da h sim cos y given 5.“. х су = 0 + tany dx Q3. solve

On integration ste - Inc HP ☺ к 1-M ina Inc I-1 (24-1) w Ť - alna talne -> (64-1) w 1-4² 22 > 2 2 -> k? 6 nba funen -> cox? y I- → 2 -cosy c² sin²y = 202 >> siny = t e (- - -) sinn y - > >> (0) h + sim (%) required explicit which B Suº