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 the derivation 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
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, ..., un such that the following is a non-homogeneous solution:
where y1, ..., yn are homogeneous solutions To find the ui'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).
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.
Some excellent practice problems including answers are available at this website. The answers are explained via video (one can also look at various screen shots to check your work as well) Note some of the integrals in these examples require formulas, but sometimes integration techniques such as u-substitution work (for example you can use u-substitution in practice problems B01, B03, and B04).
Integration Techniques:
You can now combine the homogeneous general solution with the non-homogeneous solution to find the general solution for a linear DE (where yi's are n linearly independent homogeneous solutions and Yc in a non-homogeneous solution.
Step 3 If you have initial values, plug them in to find the ci's. Note you need to plug them into the final general solution which includes any non-homogeneous part.
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. If you use g = 9.8 m/sec2, make sure you only use meters for measuring length. Similarly if you use g = 32 ft/sec2, make sure all length measurements are in feet.
All equations needed plus a homogeneous example (no external force) can be found here
Click here for a non-homogeneous example (external force)
Note when there is damping the homogeneous solution goes to 0 as t goes to infinity.
Resonance refers to obtaining a large amplitude using a small external force. For example, if there is no damping, then the homogeneous solution is of the form y = c1cos(at) + c2sin(at). Thus if you apply an external force with matching frequency (eg F = cos(at) or sin(at)), then a non-homogeneous solution is of the form y = t(Acos(at) + Bsin(at)) and thus the amplitude goes to infinity as t goes to infinity. FYI: If there is damping (which is generally the case in real life), you can use calculus to determine the largest possible amplitude when using an external force.
Theory for ch 3 and 4
Note theory is also important and can show up on this exam
Note: EVERYTHING DEPENDS ON LINEARITY.
Knowing whether or not a solution to an IVP exists and/or is unique is important, especially since in real life, one must often approximate a solution (and approximating a solution that does not exist is generally a bad idea; also, approximating one solution when there is more than one solution can be misleading).
Knowing where your solution is valid is also very important. For example, suppose your solution approximates the concentration of a drug in your blood stream. If your solution is valid for only the first 30 minutes, you do NOT want to use it to estimate drug concentration after an hour. Think about some of the direction fields you have seen. Solutions can vary dramatically depending on initial conditions.
The Wronskian is used to determine if your solutions are linearly independent and is directly related to the coefficient matrix you get when solving an initial value problem for the unknown constants. Abel's theorem is a nice shortcut for calculating the Wronskian.
Understanding how solutions are derived can help you solve other problems. For example for repeated roots and variation of parameters, you found solution by multiplying the homogeneous solution(s) by unknown function(s) and plugging in this (sum of) product(s) into the differential equation to solve for the unknown function(s). The two applications were very different: in section 3.3, we solved homogeneous differential equations whose characteristic polynomial had repeated roots. In section 3.6, we used this technique to find a non-homogeneous solution. This technique can also be used to solve one of the practice problems for midterm 2.
Section 6.1: LaPlace transform Know the definition and that fact that the LaPlace transform is a linear function.
example 1 (other examples are from later sections and thus won't be on midterm 2)