MA2051 - Ordinary Differential Equations
Test 1 Solutions - A96

A

rV l/min

B

H l/min

C

The change in the amount of water in the tank from t to is the amount added over less the amount lost. Change over . Amount added . Amount lost . Conservation requires

Divide by , take the limit as to obtain dV/dt = -rV + H. (Not required: for a complete model, add the initial condition .)

D

constant yields . Hence, there is a single steady state .

Since , this steady state represents a balance between leakage at the rate l/min and additions at the rate H l/min.

E

One stability argument: study solutions of the form V(t) = H/r + p(t), where p is small initially. Substituting in V' = -rV + H and simplifying yields p' = -rp. Since , the perturbation p decays and is stable.

Another argument: Using characteristic equations and undetermined coefficients, the general solution of V' = -rV + H is . The arbitrary constant C is determined by the initial condition. Hence, regardless of the initial state, every solution of V' = -rV + H decays to .

To find , use undetermined coefficients to guess . But solves the homogeneous equation. So revise guess to . Substitution yields A = 4.

Hence, . The initial condition u(0) = 3 forces C = 3. The solution of the IVP is .