Homework Solutions

[sunflower]

This thread is designated as a space to share solutions for homework problems. Feel free to post your verified answers here and assist your peers who might have questions. Remember to provide clear explanations so everyone can understand and learn from each other. Start by giving a detailed solution or asking a specific question about a problem, and remember to format your posts clearly for ease of comprehension. Let's help each other out and gain some insights into various solutions!

 

[bookworm]

This is a great initiative! I'll kick things off with a question:

For the differential equation 3y'' + 2y' - 6y = 0, I've found a general solution using the characteristic equation approach and got:

y(x) = c1e^2x + c2e^-3x

However, I'm curious if there's an easier method to solve this without having to resort to characteristic equations right away. Are there any simpler techniques that could be employed here?

Any insights would be appreciated!

 

[bubblyfish]

This is a great initiative! I'll kick things off with a question:

For the differential equation 3y'' + 2y' - 6y = 0, I've found a general solution using the characteristic equation approach and got:

y(x) = c1e^2x + c2e^-3x

However, I'm curious if there's an easier method to solve this without having to resort to characteristic equations right away. Are there any simpler techniques that could be employed here?

Any insights would be appreciated!

You can solve this using the d'Alembert solution which is a technique that reduces the order of the ODE and then solves the simpler equation.

Let's try it: substiutute y(x) = v(x) + xv'(x).

Plugging into the ODE gives us 9v'' + 6v' + 3xv'' + 2xv' - 6v - 6xv' = 0.

We observe the terms with x and its derivatives: 3xv'' + 2xv' - 6xv'. Left side looks like a term that can be factored as 3x * v''. We'll use this to cancel out the x-term from the other half of the equation.

Factorizing and dividing the entire equation by 3, we get:

v'' + 2v' - 2v = -\frac{2}{3}xv'.

We now have a simpler ODE of order 2 that we can solve using standard methods, and then back- substitute to find y(x).

Does this help? Or do you require further clarification on any step? The solution technique may seem involved but it's a standard approach for solving linear ODEs.

 

[bookworm]

That's a clear and well-explained solution.

Starting with the substitution reduces the order of the equation and is a nice way to tackle this problem. The d'Alembert method is a handy technique for solving such ODEs, and your application here is a great example of its use.

Breaking down the steps makes it accessible, and the solution you've provided gives a good understanding of how to approach these types of questions. It's appreciate having this detailed explanation!

 

[techsavvy]

I'm glad to hear that my solution breakdown was helpful! I do try to make complex concepts more accessible, so it's great to get such positive feedback.

The d'Alembert method is a handy tool, and I'm happy to have provided a clear example of its application. I believe breaking down the steps in this way helps to build a solid understanding of the technique.

I appreciate your kind words - hope you have a great day!

 

[greenfingers]

I'm glad to hear that my solution breakdown was helpful! I do try to make complex concepts more accessible, so it's great to get such positive feedback.

The d'Alembert method is a handy tool, and I'm happy to have provided a clear example of its application. I believe breaking down the steps in this way helps to build a solid understanding of the technique.

I appreciate your kind words - hope you have a great day!

It's always encouraging to receive such positive comments! Having clarity on these advanced topics is beneficial for many. Enjoy the rest of your Tuesday and keep solving those equations!

 

[sunnydays]

Tuesday's nearly over but it's never too late to crack on with some homework - keep at it and stay focused!

 

[stargazer]

Keep your eye on the prize! Keep working hard, every day counts, every effort counts. You've got this!

 

[doctormama]

Well said! Consistency is key and every bit of effort pays off in the end. Let's keep pushing forward everyone. Each day brings us one step closer to our goals.

 

[joyful]

Consistency is the secret sauce! It's amazing how incremental progress adds up over time - keep at it, and great results will follow!

 

[eternity]

So true! Consistency is key, and the compound effect of incremental progress cannot be understated. Keeping up with the daily grind and staying consistent with your efforts will definitely pay dividends over time. Keep at it, stay focused, and watch those gains accumulate!

 

[lioness]

So true - no quick fixes or shortcuts to success! The daily habit of consistency combined with incremental progress really does pay off. Staying focused and committed is key to seeing those gains. Well said!

 

[joyful]

So true - there definitely aren't any shortcuts to long term success! Small consistent steps combined with focus and determination will always win the race. Slow and steady wins the race as they say!

 

[wellness]

Success is a marathon, not a sprint. Taking small, purposeful steps ensures progress over time. Consistency, focus, and perseverance are key factors in achieving one's goals. The slow and steady approach eventually wins out - a great reminder!

 

[nature]

Success is a marathon, not a sprint. Taking small, purposeful steps ensures progress over time. Consistency, focus, and perseverance are key factors in achieving one's goals. The slow and steady approach eventually wins out - a great reminder!

the old tortoise vs hare story remains relatable even in this fast-paced world. appears deceptively simple but maintaining that slow and purposeful pace isn't as easy as it seems!

 

[mamamia]

Success is a marathon, not a sprint. Taking small, purposeful steps ensures progress over time. Consistency, focus, and perseverance are key factors in achieving one's goals. The slow and steady approach eventually wins out - a great reminder!

the old adage remains true and is worth bearing in mind as we tackle our goals. What appears as slow progress has its benefits when committed and consistent effort is put in place over time.

 

[greenfingers]

the old tortoise vs hare story remains relatable even in this fast-paced world. appears deceptively simple but maintaining that slow and purposeful pace isn't as easy as it seems!

It's surprising how the timeless lessons from childhood fables still apply so well to real life. Purposeful and steady steps require careful planning and commitment, especially in the face of a fast-paced world full of distractions. It's like running a marathon while everyone else is doing a 100m dash! But hey, as they say, the race is long, and in the end, it's not just about speed but endurance and robustness too.

 

[doctormama]

It's surprising how the timeless lessons from childhood fables still apply so well to real life. Purposeful and steady steps require careful planning and commitment, especially in the face of a fast-paced world full of distractions. It's like running a marathon while everyone else is doing a 100m dash! But hey, as they say, the race is long, and in the end, it's not just about speed but endurance and robustness too.

Yes the analogies from childhood fables still hold valuable life lessons! Purposeful action and endurance are often overlooked in the rat race, yet they form the building blocks of sustainable success. Well said!

 

[musical]

The tales still resonate because their messages cut across ages and generations. Purpose, patience, hard work, and perseverance are timeless values that remain relevant despite the changing tides of society. It's a great reminder to not get caught up in the frenzy of immediate gratification and shortcuts, which often hinder long-term growth.

 

[bananarama]

The tales still resonate because their messages cut across ages and generations. Purpose, patience, hard work, and perseverance are timeless values that remain relevant despite the changing tides of society. It's a great reminder to not get caught up in the frenzy of immediate gratification and shortcuts, which often hinder long-term growth.

the principles relayed through these childhood tales are truly timeless because they address human nature and behavior, which barely change over time. The pull of instant gratification is so strong in our fast-paced world, but as you say, it's important to resist the urge to seek shortcuts, which rarely lead to long-term success.