Weekly Math Problem! #47

2nd Order ODE. Sometimes, topics just come to mind and I say to myself, 🤔 "Haven't done that in a while. I should do it so I don't forget." That's pretty much it. Well, sometimes I can see the problem or parts of the problem and not quite remember the name of the topic. Then I'll have a little digging to do so that I can get the name of the topic. Well, 2nd order ode (second order ordinary differential equation) is the order for this week. 😅 Specifically, we're looking at a non-homogenous equation with constant coefficients.

⚠️Many steps involved.⚠️To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to factor a trinomial
        ✔️     How to use the method of undetermined coefficients
        ✔️     How to find derivatives

WMP! #47 says to:

Happy solving!

Check back on Friday, April 23rd for the solution, which will be posted below ⬇️.

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Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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You can't say I didn't warn you about the many steps involved in the solution to this problem. Why did I do it? Well...because it's good practice and I want to make sure that I can remember math at this level. As I challenge my myself, I challenge you. 

🧐 Let's get it!

I did my best to organize my solution and color code it so that it would be understood a little better.

The general solution to the equation can be broken down into two parts--the homogeneous solution and the particular solution.

I worked on the homogenous solution, before tackling the particular solution. Once I got bot parts, I put them together to complete the general solution to the equation.

This one was quite a bit of work but, it got done! 🤔 I think I'll do another one like this sooner than later. 

▪️ Have you ever encountered this kind of problem when learning about differential equations?

▪️ Leave your responses down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #48. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #46

Integration. I'll always find my way back to Calculus. 😉 I came across this problem on an exam review sheet. There was a note on the sheet for the students to skip this problem because, it wouldn't appear on the exam. I was intrigued so, it's appearing on the blog this week. 👩🏿‍🏫 I don't really recall solving an integral like this when I first learned integration in college. Since that was too long ago, my recollection is probably off. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to perform a substitution technique
        ✔️     How to perform completing the square method

(Do you see 👀 the hint I gave above? You're welcome. ☺️) Here is WMP! #46:

Happy solving!

Check back on Friday, April 16th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Here is my solution. I hope you can follow it. My goal is for it to be clear. I got started by dealing with the denominator.

Next, I rewrote the original problem with the completed square and saw that I can now perform u-substitution to simplify the integrand.

After completing the u-substitution, I wasn't able to integrate the function. Upon inspection, (hopefully) it can be seen that trigonometric substitution needs to be used to complete the integration.

Ahhh...integration is complete...but the process isn't complete. 🤨 Conversions need to occur so that the answer can in term of x to match the original problem. Plus, I need to figure out what cos 𝜃 is equal to.

Now, the process is complete! 🥳 Let me know what you did differently, or if you didn't do anything differently.

▪️ Leave your responses down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #47. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem! #45

Diagonalization. I want to go over some linear algebra this week so, here is a problem to help me with that. I landed on diagonalization of a matrix because, for some reason, it stuck with me. I can't really explain why. Check out my other WMPs that involved working with matrices--WMP! #4 and WMP! #14. I don't have much else to say (due to tiiirrreeeddd-ness) 😩. 

⚠️Many steps involved.⚠️To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to perform row operations
        ✔️     How to find the determinant of a matrix
        ✔️     How to solve a quadratic equation
        ✔️     How to multiply matrices
        ✔️     How to find the inverse of a matrix

All calculations can be done without a calculator but, feel free to use a matrix calculator to check your work. Ready?? 😁🥴 WMP! #45 says...

Happy solving!

Check back on Friday, April 9th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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I'm just gonna go 'head and get into this calculation...computation...situation...

😅

Step 1. Find the Characteristic Polynomial

Step 2. Find the Eigenvalues 

Step 3. Find the Eigenspaces and Eigenvectors 

Step 4. Define Invertible Matrix, F, and Diagonal Matrix, D 

Step 5. Complete the Diagonalization

 

(Check the Diagonalization)

Step 6. Complete the Diagonalization

(Final Check)

**This plot was created using Desmos' Matrix Calculator.

WHEW!!! That was a lot. (You can't say I didn't warn you. With that...I think I'm gonna take it easy next week. 🥴)

▪️ Did you survive this calculation?

▪️ Have you ever taken Linear Algebra?

▪️ Leave your responses down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #46.

Cheers!

The Younge Lady