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IVPs: Consider the diï¬€erential equation dy

dt =t

t2y+y.

(1) Find the general solution for this DE.

(2) Find the solution for this DE that passes though the point (1,2).

Picardâ€™s Theorem: T/F + Explanation: Both conditions of Picardâ€™s Theorem hold for the following IVP:

dv

dt =kv2/3,

v(3) = 4.

Linear Operators: Show that L(y) := yâ€²+ty2fails to be a linear operator.

Mixing: A tank initially contains 2000 gallons of water containing 400 pounds of salt. Fresh water is pumped

into the tank at the rate of 50 gallons per minute and a well-mixed solution is being drained from the tank

at the rate of 100 gallons per minute. Determine the number of pounds of salt remaining in the tank when

the tank holds 1000 gallons.

Stability & Equilibria: Consider the Predator-Prey model given by

dR

dt = 2R(1 âˆ’R)âˆ’RB,

dB

dt =âˆ’B+RB

2.

Determine and plot the nullclines and equilibrium points for the system.

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