4. y’ =
x2 — 2
e-v
5. 3×2(l + ln ||/|) + ^ – 2y’j y’ = 0
6. Solve the initial value problem: ty’ – 2y = 512, t > 0, i/(l) = 2.
7. Solve the initial value problem: y’ – 2xycos(x2), t/(0) = 2. Write y explicitly as a function of x.
8. A large tank is partially filled with 1(K) gallons of water in which 10 pounds of salt is dissolved. Brine containing 5 pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The stirred solution Is then drained at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes.
9. Solve the second order equation ty” — xf = 312 by reducing it to a first order equation. That is, make the substitution v = y solve for v, and then integrate to find y.
10. Consider the initial value problem: y = 2^, y(0) = 0.
a. Show that this initial value problem has more than one solution (in fact, it has infinitely many, but you only need to find two).
b. Why doesn’t this contradict the theorem on existence and uniqueness (2.4.2)?
5. Consider a circuit with inductance L, resistance /?, and emf E which are all constant. Then Kirchoff’s law states that the current I(t) satisfies