Can you please explain this correctly?

MC: 3 5 FR : 4 NM AP Calculus AB Name Tavel lopez NIT 7 EXAM - Differential Equations FORM A A bacteria colony with population y grows according to the differential equations dy = 0.4y. There are 2000 dt bacteria initially. Find the approximate number of bacteria at time t = 7. (A) 32,889 (B) 14,000 (C) 16,446 (D) 20,885 . The differential equation da = 12 9-3q in separable form is: dt dq (A) (+ 2 - 3 ) dt (B) dy = (12q-39) dt (C) - da + 39 =12 dt (D) da = (12 - 3 ) dt A student attempted to solve the differential equation - dy = xy with initial condition y = 2 when dx x=0. In which step, if any, does an error first appear? Step 1: [ - dy = ] x dx Step 2: In y = 2 1 + C $ 2 Step 3: |yl = e2 + C Step 4: Since y =2 when x = 0, 2 =e +C Step 5: y= e2 +1 (A) Step 2 (B) Step 3 (C) Step 4 (D) Step 5(B )y=els Which of the following is a solution to the differential equ (4) y = 20' SOM (C) y =sin(2x) Shown is a slope field for which of the following differential equations? A) ay y - 2 dy - 2 (B) x+1 dx dy dy y + 2 1 1 1 1 x +1 x+1 According to Newton's Law of cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. Suppose a corpse at a temperature of 32 C arrives at a mortuary where the temperature is kept at 10 C. Then the differential equation satisfying the temperature T of the corpse t hours later is (A) dI = KT(T- 10) B) dl =KT(T- 32) dt dt dT (C) al = k(T-32) (D L = k(T-10) dt dtWhich of the following is a solution to the differential equation y" -4y =0? (A) y = 2e* (B) y = e2x (C) y = sin(2x) (D) y = cos(2x) Find the particular solution to the differential equation ay = 4X when y ( 2 ) = - 2 . dy 42 dy = 4x dy y Jilly = juxdx enly ! = 2* Find the particular solution if day = - = ysec x and y = 5 when x = 0. dy U = sec 2 = y sec ( x ) . du do aof- dy = ten day dy = sec x lx y y dy = see Exlodx Sxy 9. Consider the differential equation given by dy 2 On the axes provided to the right, sketch a slope field for the given differential equation at the nine points indicated. 2 Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) =3 Use your solution to find f(0.2). dy dx To At each point (x, y) on a certain curve, the slope of the curve is -. If the curve contains the point (0,8). 2Vx then what is the equation of this curve