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APMA 0350 − HOMEWORK 1
Problem 1: (4 = 2 + 2 points)
(a) For which values of r is y = e rt a solution of y ′′ + 3y ′ − 10y = 0 ?
(b) Determine whether y = t 4 is a solution of t 3 y ′′ + t y′ − 8y = 0
Problem 2: (5 points, 1 point each) Find the order of each equation and state if it is linear or non-linear. If it’s linear, state if it is homogeneous or inhomogeneous. No justification needed
(a) e t y ′′ + y ′ − t 2 y = 13t 3
(b) (y ′ ) 2 + t y + cos(y) = 0
(c) y ′′′ + 10y ′′ + 5y ′ − 2y = 0
(d) cos(t) y ′ + t y = 0
(e) y ′ = t 3 y 2 + t y
Problem 3: (3 points) Solve y ′ =2y + 3 y(0) =3
Hint: To do this, please divide both sides by 2y + 3 and recognize the left-hand-side as a derivative, just like we did in lecture with ln.
Problem 4: (4 = 2 + 2 points) (Mini-Theory) Consider the ODE
y ′′ − 4y ′ + 4y = 0
(a) Show that y = Ae2t +Bte2t solves the ODE, where A and B are constants
(b) Show that y = Ae2t + Bte2t is the only solution to the ODE
Hint: Consider e −2t y ′′ similar to what we did in lecture
Problem 5: (4 points) (Application) You’re studying the growth of a bacteria population and you’re noticing that the rate of change of the population is equal to three times the number of bacteria present. Set up an ODE for the the number of bacteria and calculate the amount of time it takes for the population to double. Does the answer you found depend on the initial population y0 of the bacteria?
Note: Direction fields will be part of the programming assignment