Step 1: Solve homogeneous equation

See page 1 of sections 3.1, 3, 4 as well as page 2 for examples. Remaining part of this handout includes (i) an explanation as to why the exponential function is a good guess for linear homogeneous differential equation with constant coefficients and (ii) shows the derivation for simplifying the solution when roots are complex and when roots are repeated. A handout on 3.3 shows why we use Euler's formula to simplify the solution when the roots are complex.

Note you must be able to factor.

Step 2: Find one non-homogeneous solution (when the DE is linear non-homogeneous)

We covered 2 methods. The first method, section 3.5: undetermined coefficients involves making an educated guess and solving for the undetermined coefficients. Note you can use this method if you can think of a good guess that you can plug into the RHS to get out the LHS. For example, if g(t) is one of the following

If g(t) is a product of the above, you can also use this eduated guessing method (by guessing a product of these).

This 3.5 handout shows what happens when you guess wrong and how to figure out the correct guess.

This 3.5 worksheet is good practice for guessing non-homogeneous solution

I highly recommend reading over the answers to the above linked worksheet as the worksheet answers contain detailed explanations.

In the second method, section 3.6 variation of parameters, we showed that for a second order linear DE, we can find u₁ and u₂ such that the following is a non-homogeneous solution:

where y₁ and y₂ are homogeneous solutions

For an nth order linear DE, we can find u₁, ..., u_n such that the following is a non-homogeneous solution:

where y₁, ..., y_n are homogeneous solutions To find the u_i's, you can either plug in the above into your DE and solve for them (but you only have one equation, so you can choose other equations) or you can use the formula (which came from using Cramer's rule and thus involves the Wronskian).

This handout gives background including determininants, Cramer's rule, the formula and an example for the 2nd order case

These notes cover an order 3 example

Another order 3 example from Paul's online notes The derivation for the order 2 case is included here

You can now combine the homogeneous general solution with the non-homogeneous solution to find the general solution for a linear DE (where y_i's are n linearly independent homogeneous solutions and Y_c in a non-homogeneous solution.

Step 3 If you have initial values, plug them in to find the c_i's. Note you need to plug them into the final general solution which includes any non-homogeneous part.

A 3.5 IVP example.

Mechanical Vibrations

Make sure you can set up the IVP including both the differential equation and initial values. Remember our convention that the positive direction is down.

All equations needed plus a homogeneous example (no external force) can be found here

Click here for a non-homogeneous example (external force)

More application examples

Theory for ch 3 and 4

Note theory is also important and can show up on this exam

Section 6.1: LaPlace transform Know the definition and that fact that the LaPlace transform is a linear function.

6.1 examples

example 1 (other examples are from later sections and thus won't be on midterm 2)