I’ve finished grading Exam #2, and I incorporated your scores into the Midsemester Grade. You can find your Midsemester Grade on the Grades page (click the link in the menu above) – if you forget the password for the Grades page, send me an email or ask a fellow student. The Midsemester Grade contains detailed scores for all the WeBWorK and OpenLab assignments to date, plus the first two exams – you can find your Exam #2 grade in the Midsemester Grade breakdown.
Let me know if you have any questions,
Prof. Reitz
An excellent question that I received in email today with regards to WeBWorK #9:
Hi professor Reitz, on problem number two for the new homework, when I try to solve for the particular solution, everything on the left side cancels.
This will happen when the expression on the right side of the equation also happens to be one of the solutions to the homogeneous equation. We deal with it in much the same way we dealt with repeated roots in homogeneous equations: When guessing the particular solution to the nonhomogeneous equation, multiply your guess by (for example, use instead of . Here’s an example.
Example:
The general solution to the associated homogeneous equation is:
General solution:
Notice that one of the basic solutions involves , which matches the right hand side of the original equation. Because of this, we would make the following guess for a particular solution:
Guess:
Notice that when you take the derivative, you will still end up with a term involving just (without the extra “t”), which will allow the left hand side of the equation to equal the on the right side.
Let me know if you have any questions (post a comment!),
Prof. Reitz
UPDATE: I added one problem (#5) to the Exam Review, from the topic exact equations — you may remember that we covered this in class just before the first exam, but it was not on the first exam.
Hi everyone,
The review sheet for Exam #2 (which will take place on Thursday, April 2nd) is posted on the Handouts page. Let me know if you have any questions.
Here are some examples of Euler’s Method problems for you to try. These are for practice only, they will not be collected. NOTE: The first example is the one we solved in class on Thursday – I’m including it here for your reference.
Questions? Problems? Leave a comment on this post and I’ll respond here.
Example 1. Given the initial value problem with , approximate the value of . Use 5 steps.
Example 2. Consider . Suppose . What is ? Use a step size of .
Example 3. If , use Euler’s method with 4 steps to approximate for the particular solution satisfying
Example 4. Given with , estimate the value of using 6 steps.
Solutions to these examples are provided here for your reference – if you notice an error, please let me know by leaving a comment below (or sending an email).
Exam #1 will take place on Thursday, March 5th, during class. The review sheet for Exam #1 has been posted on the “Handouts” page. Let me know if you have any questions.
Shortcuts are dangerous things – they may save you time, but they usually don’t help you understand the problem. Because of this, it’s usually important to have a thorough grasp of the basic idea of how to solve a problem before learning the shortcut. Since you’ve had a few days to wrestle with the “Integrating Factors” problem, I wanted to share a standard shortcut (covered in the text, but not yet discussed in class) for solving these problems, which condenses much of the algebra into two formulas. You are welcome to use it, or not, as you prefer.
Shortcut for solving Integrating Factors problems:
Step 1: Rewrite the differential equation in the standard form:
In practice, this usually just means getting the and on the same side, and dividing to get rid of anything in front of the .
Step 2: Find , by plugging in:
That is, integrate the function in front of , and then raise to the power of the result. This gives .
Step 3: Multiply both sides of the equation by , and then integrate both sides. Notice that the integral of the left side will always equal .
Step 4: Finally, solve the resulting equation for y. You’re done!
Special Bonus Shortcut II: The work of Steps 3 and 4 can be condensed into the following formula, which can be used to find directly after completing Step 2:
That is, multiply by the function from the right hand side of the differential equation, integrate, and multiply the result by $\frac{1}{\mu}$.
NOTE: The standard form mentioned in Step 1 shows up a lot – in fact, even if you are not using the shortcut formulas above, it is considered “pretty standard” to rewrite your equation in standard form before solving the problem.
An example using the Shortcut: NOTE: In the video, he uses x as the independent variable, instead of t.
https://www.youtube.com/watch?v=Et4Y41ZNyao
Another example: NOTE: Towards the end of this example, when integrating the right-hand-side, he uses integral of , which is — if this looks unfamiliar, you should review the derivatives of the inverse trig functions
Falling objects. This video starts with some discussion of where the differential equation comes from, then carries out a problem from start to finish. A couple of things to note:
1. The problem uses U.S. Customary Units (feet, pounds, etc.) – in particular, acceleration due to gravity is 32 ft/sec^2 (instead of 9.8 m/sec^2).
2. In solving the example, the differential equation is solved using the Integrating Factor method (which we will learn on Thursday).
3. The “limiting velocity” is just another term for the terminal velocity.
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