Solving linear differential equations:

Step 1: Solve homogeneous equation

See page 1 of sections 3.1, 3, 4 as well as page 2 for examples. Pages 1 and 2 are the most relevant. The 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, (ii) shows the derivation for simplifying the solution when roots are complex and the derivation when roots are repeated, (iii) Some theory including Wronskian.

A example from 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 (section 4.2). See also the 3.5 examples that will be posted later.

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.

I'll post more examples from before class later.

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.

NOTE: If you use variation of parameters and cannot do the integrals when you should have used undetermined coefficients, do NOT expect much partial credit. Similarly, if you get the formula wrong or can't do the integrals,this will also affect partial credit. Thus you might want to use Cramer's rule to solve Wu' = b for more room for partial credit as illustrated here.

Integration Techniques:

• Integration by substition
• Integration by parts
• Integration by partial fractions
Note: Partial fractions are also used in ch 6 for a different application.

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. 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)

Some graphs from our textbook

More application examples

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.

Updated handout with formulas including trig identities, discussion of damping, transient vs steady state solution, resonance

Quiz 3 including answers. Note, I gave different versions so that you can compare differences in wording, etc. For 3.7 and 3.8, make sure your units are consistent (for example use kg for mass and meters for length).

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.

See simple example 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.

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

6.1 examples

example 1

You must memorize formulas 1-6, 9-11,14, 18-19 from the LaPlace table. First page is the LaPlace transform table. Page 2 summarizes useful algebraic techniques.

Note you can use formula 14 to derive formulas 2, 9, 10, 11 from formulas 1, 5, 6, 3, respectively.

Section 6.2: Use LaPlace transform to solve linear differential equation 6.2 example

Some additional notes: Derivation of formula 18