Question

(1) Find the general solution of dy/dt=y

(2) Find the solution y = f(t) of dy/dt = t/y^2 such that f(1) = 2.
(3) Find the general solution of dy/dt = t³y. Check for yourself if you got the correct family by plotting a sample solution together with a slope field.
(4) Find the general solution of the logistic differential equation: dy dt = y(2-y). Using slope field analysis (and possibly a computer generated slope field) explain why you are confident that your family of solutions is correct?
(5) A discharged capacitor C = 2700F is connected in a series with the resistor R = 1 Omega . (a) Find the general formula for the voltage drop V_(cap) = f(t) across the capacitor if the circuit is connected at time t = 0 to a battery supplying voltage V_(bat) = 2.7V. (b) How long does it take for the voltage accross the capacitor to reach 1.45V? The differential equation governing the voltage drop on the capacitor in this circuit is:​

Answer 1

1) dy/dt = y has the general solution y = Ce^t where C is any constant.

2) We need to find f(t) such that f'(t)= t/f(t)^2 and f(1) = 2. The solution is f(t) = sqrt(t^2 + 4).

3) The general solution to dy/dt = t^3y is y = Ce^{t^4/4}

4) The general solution to the logistic equation is y = 1/(1+Ce^(-2t)) which describes

S-shaped curves where y approaches 1 as t approaches infinity.

5) (a) V_cap = V_bat(1-e^(-t/RC)) where R is the resistance and C is the capacitance.

(b) Plugging in V_cap = 1.45 and V_bat = 2.7 into the equation, and solving for t yields

t = -270ln(0.535) = 416 seconds