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AMATH 250
Examination Final
Spring 2024
1. Find the solutions (implicit or explicit) to the following equations.
(a) 
[Hint: Use an appropriate change of variable].
(c) 
(d) 
2. (a) Consider the initial value problem
Using the existence and uniqueness theorem, what can you predict about the solution of the IVP?
(b) Find the Wronskian of two solutions of the DE.
[Hint: You do not need to solve the DE. Your solution should be written as W(y1, y2)(t) = c f(t), where f(t) needs to be determined.]
3. (a) Use the Laplace Transform to solve the integral equation.
(b) Solve the initial value problem,
where δ(t) is the Dirac delta function.
4. Consider the vector differential equation 
(a) Find the general solution to the vector differential equation.
(b) Sketch the phase portrait of the solutions. Include any exceptional solu-tions and isoclines.
5. Using the method of variation of parameters, find a particular solution to the system:
The solution to the homogeneous equation is
6. (a) Consider the following DE for the population N(t):
Here r and K are the intrinsic growth rate and carrying capacity, re-spectively, and A and B are constants. Use 
and 
to nondimensionalize the IVP.
(b) When a fluid in a pipe accelerates linearly from rest, it initially exhibits laminar flow, which then transitions to turbulence at a specific time, de-noted as ttr (seconds). This transition time depends on several physical parameters: the pipe diameter D (cm), the fluid’s acceleration a (cm/s2 ), density ρ (g/cm3 ), and viscosity µ (g/(cm.s)). Using the Buckingham Pi theorem, determine the relationship between ttr and the other mentioned physical parameters.