MAT 330 Final Exam

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MAT 330 Final Exam

Assignment Instructions: For each problem, be sure to show all steps for arriving at your solution. Work within this document. Use an equation tool as needed, and submit everything in one file.

  1. For the below ordinary differential equation, state the order and determine if the equation is linear or nonlinear. Then find the general solution of the ordinary differential equation. Verify your solution.

2. For the below ordinary differential equation, state the order and determine if the equation is linear or nonlinear. Then find the general solution of the ordinary differential equation. Verify your solution.

3. For the below ordinary differential equation with initial conditions, state the order and determine if the equation is linear or nonlinear. Then find the solution of the ordinary differential equation, and apply the initial conditions. Verify your solution.

4. For the below ordinary differential equation, state the order and determine if the equation is linear or nonlinear. Show the equation is exact and then use the method for exact equations to find the general solution of the ordinary differential equation. Verify your solution.

5. For the below ordinary differential equation with initial conditions, state the order and determine if the equation is linear or nonlinear. Then find the solution of the ordinary differential equation, and apply the initial conditions. Verify your solution.

  • Find a solution to  using variation of parameters.
  • Find the solution to the differential equation in problem 6, this time using the method of undetermined coefficients.
  • A baked cake is taken out of a 350o oven and placed on a table in a 70oF room. It cools according to Newton’s law of cooling. Ten minutes later the temperature is 315oF.  What is the temperature of the cake after 30 minutes?
  • Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor, , an inductor, , and a capacitor,  in an electrical circuit must be the same as the voltage source, , applied to that  circuit. Applying the additional fact that the current, , is related to the charge,  , by the relationship ,  the resulting ODE model for the charge in a circuit is:

If a source is connected to an RLC circuit with a  henry inductor, L,  a 1 ohm resistor, R , and a   farad capacitor, C, find the charge   given that  and .

  1. Using the Newton’s second law model for a vibrating mass-spring system with damping and no forcing,

,

find the equation of motion if  kg,  kg/sec,  kg/sec2, , and  m/sec.