Part I: Second Order Homogeneous Differential Equations 1. What is meant by homo

Question

Part I: Second Order Homogeneous Differential Equations

1. What is meant by homogeneous? What should you look for to determine the order of a differential equation?




2. Given y^''-4y^'-12y=0 replace y^'' with m^2, y^' with m, and y with 1 to create the characteristic equation.

Characteristic equation is ______________________________



3. Solve the characteristic equation above for the roots m_1 and m_2. Hint: you may need to solve by factoring or by using the quadratic formula.





4. Write the solution of the differential equation based on the roots found in question 3.

a) Are m_1 and m_2 real numbers and not equal to each the other? Then the solution is
y=C_1 e^(m_1 x)+C_2 e^(m_2 x)

b) Are m_1 and m_2 real numbers and equal to each the other? Then the solution is
y=C_1 e^(m_1 x)+C_2 ?xe?^(m_2 x)

c) Are m_1 and m_2 complex numbers of the form a + bi and a - bi? Then the solution is
y=C_1 e^ax sin?(bx)+C_2 e^ax cos?(bx)


Solution is _____________________

5. Solve the following homogeneous differential equation by repeating the process above.

3y^''+5y^'+7y=0

Create the characteristic equation.




Solve the characteristic equation.





Write the solution of the differential equation.




Solution is_________________________

6. Solve the following homogeneous differential equation by repeating the process above.

16y^''+40y^'+25y=0

Create the characteristic equation.




Solve the characteristic equation.





Write the solution of the differential equation.




Solution is_________________________


7. Take the result of question 6 and find the particular solution given that y(0) = 1 and y^' (0)=0 . Why do we need two initial conditions for second order differential equations? How are the initial conditions used to find a particular solution? What must be done in order to use the second initial condition?











Particular Solution is___________________________

Explanation / Answer

Given y^''-4y^'-12y=0 replace y^'' with m^2, y^' with m, and y with 1 to create the characteristic equation is m^2 -4m-12 =0 .

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