Question

1. Solve the following homogeneous differential equation. ty' = 1. cos (6) + y 2. Solve the following Bernoulli differential equation 3. Solve the following initial value problem. (Hint: transform the equation to a separable equation through a substitution) y-(x + y + 1)? (0) - V3 - 1 4. Let T represent the temperature (in °F) of an object in a room whose temperature is kept at a constant 60°. If the object cools from 100 to 90° in 10 minutes, how much longer will it take for its temperature to decrease to 80°?

Answer 1

Anar aly' = a conly) + y Y'- conly) + Lot ug - yaux - Now , doya 1 + & + 2 do = con us to Integrating on both sides. log [secl + ban luj] = Indstine Sec (W) + ton (W) = ac Sec (4) + tan ) = 20 → 2 1 + 2 ) ay = fall = ₃ [lny-in 12-y)] = lnxt lhe - Infy) = ln (xc? y = x 2 Y = 20x 03² - 017² ) y = 206032 ly 1+ 100012 Any

Answer 2

Day' = a cos (12) ty > y = cos (1/2) +(3/2) 1. let y=vx . Ca - 1+x) => WEn dat = cos v+v => x dve y = cose > log/sec vitanu/= ln xtc -> Sec vt tan ve Enes 1 sec (2) + tan (A) = x C1 are constants Where C & C @ y'= - + 2 y = £ (24-43) → des c 4 (2-4) => 1 S ( + 1/4 dy = lnx 키 Mg(24)=he → son ) zen Xi ( Some constant Scanned with CamScanner

du ~ y'= (x+y+02 400) = 13-1 let x+y+1 = t d+ d y = det har -pulting in differential egy -1+ dietet = 42 → At = 1++ Cast 2 fax => a=tan'ttc a a= tant (a+y+1)+c at azo, y(0)=13-1 => 0 = tañ' (0+13++8)+C -> c=-tant => fc = -1.047 :: 2 = tant (x+y +1–1.047 > 18 = tan (X+1.047) –3-17 6 Newton's law of cooling dTs-k{t-Tae) = far -fkat - THE TR + (--) ** Scanned with CamScanner

at To = 60°F, To = 100 TH=90 @ t= 10 min putting in equi 90=6079 90 = 60+ (10060) é R 10 To = è Rio Now, ow, from goor to 80F To : 80 = 60+(90-60) ēki Rt → 20 zert 8o llom den ► en (26) = inte en la) to so en > t = 4:09 min Scanned with CamScanner