Weekly Math Problem #75

1st Order ODE. This week's problem type was inspired by a student who just wanted to review and be more familiar with differential equations (of the ordinary type). I rarely get a student who needs assistance with differential equations, so you know I was excited to help the student. 🤩 These types of sessions are great for me because they force me to remember concepts and topics I don't use often. It also allows me to stay sharp. 🔪

Check out WMP #2 and WMP #47 for other problems involving differential equations. 

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

        ✔️     How to solve a 1st order, linear differential equation        

             

WMP! #75 wants us to ...

Happy solving!

Check back on Saturday, February 26th 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! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

If you've checked out my previous differential equation WMPs, then you're aware that solutions to the problems are not short. 😒 Nonetheless, here we go... 

Once it's been identified that we're dealing with a first order, linear ordinary differential equation, it need to be put in a form that will allow us to move forward with solving it. Make the coefficient of dy/dt 1, then identify p(t) and g(t). 

Now that p(t) has been identified, we can now find the integrating factor.

The integrating factor is crucial in figuring out what the function is. 

Figuring out the function is great, but we're not done yet. We've got to deal with the constant. Now, we'll use the condition provided in the problem.

We have a function, and we have the value of the constant, C, that will yield the specified condition. Let's put it together.

And before we go, let's take a quick look at the function we just found:

** This plot was generated using Geogebra.org's calculator. **

▪️ Were you able to solve for y?
▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. ✏️
We're on to WMP! #76. 🤓

Cheers!

The Younge Lady

Weekly Math Problem #51

Differentiation. Coming back to a calculus classic this week--the derivative! I feel like calculus helps my mind to feel nice. 😌

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

        ✔️     How to find the derivative of a polynomial

WMP! #51 says...

Happy solving!

Check back on Friday, May 28th 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! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

We've been asked to find the derivative of a polynomial written in product form. This lends itself very easily to the product rule for differentiation. (Also, this is the way to do it because, expanding the polynomial is too time consuming. 😩) This requires us to identify the factors of the polynomial so that we can tag them accordingly. The formula below has f and g in it but, other identifiers can be used. After identifying the factors, I find the derivative for each one, then plug in all components into the formula.

As soon as you complete the substitution, you have found the derivative. The remaining work that's done is for simplification and better presentation of the derivative. 

▪️ Were you able to find the derivative??

▪️ Let me know what you thought about this week's problem in the comments section.

Thanks for solving with me this week!

Up next, WMP! #51. 👍🏿

Cheers!

The Younge Lady

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 ⬇️.

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

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

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! #2

Ordinary Differential Equations (ODE). For some reason, differential equations was on my mind, even when this blog was just a thought. I couldn't help thinking about a notebook 📓 that I did some ode work in years ago. I knew where the notebook was and located the specific problem that kept coming to mind. Let me tell you...when I flipped those pages, the dates were from 2010/2011. 🔟 years ago! Whoa! 😲

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

     ✔️  How to integrate polynomial functions 
     ✔️  Completing the square technique 
     ✔️  Solving higher order polynomial equations

Here goes WMP #2! 

Happy solving! Check back on Friday, May 15th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Based on the nature of the derivative in the problem, the separation of variables method was used:

In finding the domain, I used both methods. What I'll do is show how to use the cubic formula to find the domain in a separate blog post. So, keep an 👁 out for that one. I knew the cubic formula existed but never used it. This week's problem was a great opportunity for that.

This solution had quite a few steps--separate the variables, integrate both functions, use the initial condition to find the constant, isolate y to find the explicit solution, then for accuracy, find the domain of that function. 🥴 *wipes perspiration from forehead* Despite the work, I do like this problem because different types of math were needed to solve in completion.

◾️ Did you arrive at the same answer I have above?? 

◾️ When was the last time you solved an ordinary differential equation??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Cheers!

The Younge Lady