Engineering High School

It is urgent! Can you solve this equation by power series method?

dy/dx=[tex]0.2x^{2}+y[/tex]

I have 35min to deliver the answer and I don't know how to do it.

Answers

Answer 1

Answer:

yes

Explanation:

Yes, I can solve this equation by power series method. Here are the steps:

Assume a power series solution of the form. [tex]y = \sum_{n=0}^{\infty} a_n x^n[/tex]Differentiate term by term to get [tex]y' = \sum_{n=1}^{\infty} n a_n x^{n-1}[/tex]Substitute into the equation and simplify to get [tex]$$\sum_{n=1}^{\infty} n a_n x^{n-1} = 2 \sum_{n=0}^{\infty} a_n x^{n+2} + \sum_{n=0}^{\infty} a_n x^n$$[/tex]

Re-index the sums to have the same power of x and combine them to get                                              [tex]$$\sum_{n=0}^{\infty} [(n+1) a_{n+1} - 2 a_n x^2 - a_n] x^n = 0$$[/tex]

Equate the coefficients of each power of x to zero and solve for the recurrence relation [tex]$$a_{n+1} = \frac{2 a_n x^2 + a_n}{n+1}$$[/tex]

Use the initial conditions [tex]$y(0) = a_0$[/tex] and [tex]$y'(0) = a_1$[/tex] to find the values of [tex]$a_0$[/tex] and [tex]$a_1$[/tex]

Substitute the values of [tex]$a_0$[/tex] and [tex]$a_1$[/tex] into the recurrence relation and find the values of [tex]a_2[/tex],[tex]a_3[/tex], etc.

Write the solution as [tex]$$y = \sum_{n=0}^{\infty} a_n x^n$$[/tex]

[tex]For example, if we have $y(0) = 1$ and $y'(0) = 2$, then we get $a_0 = 1$ and $a_1 = 2$. Then we can find $a_2$, $a_3$, etc. by using the recurrence relation:a_2 = \frac{2 a_1 x^2 + a_1}{2} = \frac{5}{2}x^2a_3 = \frac{2 a_2 x^2 + a_2}{3} = \frac{25}{12}x^4a_4 = \frac{2 a_3 x^2 + a_3}{4} = \frac{125}{96}x^6[/tex]

The solution is then [tex]y = 1 + 2x + \frac{5}{2} x^{2} + \frac{25}{12} x^{4} +\frac{125}{96} x^{6}+...[/tex]

Answer 2
To solve the differential equation using the power series method, we can assume a power series representation for the function \(y(x)\) as:

\[y(x) = \sum_{n=0}^{\infty} a_n x^n\]

Let's differentiate this series with respect to \(x\):

\[\frac{dy}{dx} = \sum_{n=0}^{\infty} a_n n x^{n-1} = \sum_{n=0}^{\infty} a_n (n+1) x^n\]

Substituting this into the given differential equation, we get:

\[\sum_{n=0}^{\infty} a_n (n+1) x^n = 0.2x^2 + \sum_{n=0}^{\infty} a_n x^n\]

Comparing the coefficients of like powers of \(x\) on both sides, we have:

For the left side:
\(a_0\) term: \(a_1 = 0.2a_0\)
\(a_1\) term: \(2a_2 = 0.2a_1 + a_0\)
\(a_2\) term: \(3a_3 = 0.2a_2 + a_1\)

And so on. We can use these recurrence relations to find the values of the coefficients \(a_n\) one by one.

To get started, let's determine the first few coefficients:

\(a_1 = 0.2a_0\)

\(2a_2 = 0.2a_1 + a_0\)

\(3a_3 = 0.2a_2 + a_1\)

Once we have determined the values of \(a_0\), \(a_1\), \(a_2\), and \(a_3\), we can continue the process to find more coefficients using the recurrence relations.

I recommend using a computer algebra system or a software package such as MATLAB or Mathematica to automate this process and calculate the coefficients efficiently. It might be challenging to complete this calculation manually within the given time frame of 35 minutes.

Related Questions

What is the first thing you should do after retrieving a boat onto a trailer?

A.) Transfer all gear from the boat to the vehicle
B.) Check that the trailer lights are working
C.) Secure any items that are loose in the boat
D.) Pull the trailer well away from the boat ramp

Answers

Answer:

D.) Pull the trailer well away from the boat ramp

Explanation:

After retrieving a boat onto a trailer, the first thing you should do is move the trailer away from the boat ramp to allow other boaters to use the ramp. This helps maintain a smooth flow of traffic and prevents congestion. Once the trailer is in a safe location, you can then secure the boat to the trailer using straps or tie-downs. After securing the boat, you can proceed with other tasks such as transferring gear from the boat to the vehicle, checking trailer lights, and securing any loose items within the boat.

The first thing you should do after retrieving a boat onto a trailer is to pull the trailer well away from the boat ramp. Therefore option D is correct.

After retrieving a boat onto a trailer, the first thing you should do is pull the trailer well away from the boat ramp. This is important for several reasons:

1. Safety: Pulling the trailer away from the boat ramp ensures that you are not blocking the ramp, allowing other boaters to access the water. It helps maintain a smooth flow of traffic and prevents congestion and delays at the ramp.

2. Courtesy: By promptly moving the trailer away from the boat ramp, you show consideration for other boaters who may be waiting to launch or retrieve their boats. It is good boating etiquette to minimize the time spent at the ramp to allow others to use it efficiently.

3. Parking: Moving the trailer away from the ramp provides you with the opportunity to find a suitable parking spot for your trailer and vehicle.

It allows you to safely and securely park your trailer in an appropriate designated area, ensuring it is not obstructing traffic or creating any hazards.

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A company has the goal of developing technology to remove carbon dioxide
from the atmosphere. Its design team is making a model based on one
possible solution to the problem. Which step of the engineering design
process comes next?

Answers

Answer: The next step in the engineering design process would be to create a prototype of the model and test it to see how well it works.

What was the most technologically advanced car in the 90's

Answers

Answer:

There were several technologically advanced cars in the 1990s, including the Mitsubishi 3000GT, Porsche 959, and Williams FW15C Formula One car. It is difficult to determine a single car as the "most technologically advanced" since advancements varied across different models and manufacturers. However, the 1993 Williams FW15C is often considered one of the most technologically advanced Formula One cars of all time , featuring advanced electronics, active suspension, and advanced aerodynamics. Additionally, the 90s were a golden decade for technologically advanced Japanese cars , including iconic models like the Nissan Skyline GT-R and Toyota Supra.

Explanation:

Answer:

While there were many advanced cars in the 90's the McLaren F1 was one of the most technologically advanced cars in the '90s.

Hope this helps:))

What is the proper technique for anchoring?

A.) From the bow
B.) Over the port side
C.) Over the stern
D.) From the starboard quarter

Answers

From the Bow

Head slowly into the wind or current to a position upwind it up current of where you actually want to end up

Which of the following is recommended as part of the annual maintenance program for a gasoline-powered boat?

A) Change the spark plugs with automotive plugs and check spark distance
B.) Drain and check bilge along with topping off the anti-freeze level
C.) Calibration and re-installation of an automotive-type fuel pump
D.) Examination of thru-hull fittings for signs of leakage or corrosion

Answers

The most recommended choice as part of the annual maintenance program for a gasoline-powered boat would be option D) Examination of thru-hull fittings for signs of leakage or corrosion.

Thru-hull fittings are essential components of a boat's plumbing system.

They are responsible for allowing water to enter or exit the boat for various purposes such as cooling, bilge pumping, or livewell circulation.

Regular inspection of thru-hull fittings is crucial to ensure their integrity and functionality.

Examining thru-hull fittings for signs of leakage or corrosion is important for several reasons.

Firstly, leaks in thru-hull fittings can lead to water ingress, which can cause damage to the hull, electrical systems, or equipment onboard.

Detecting leaks early on can help prevent further damage and potential sinking of the boat.

Secondly, corrosion can weaken the fittings over time, compromising their structural integrity.

Corroded thru-hull fittings may fail, leading to water intrusion or even loss of the fitting itself.

By inspecting for signs of corrosion, such as rust or deterioration, necessary maintenance or replacement can be planned to ensure the fittings are in good condition.

Regular examination of thru-hull fittings should include checking for tightness, cracks, wear, or other visible damage.

It is also advisable to ensure the proper operation of any valves associated with the fittings.

While the other options may also be part of a maintenance program, examining thru-hull fittings for leakage or corrosion is particularly crucial for the safety and reliability of a gasoline-powered boat.

It helps mitigate potential risks associated with water ingress, hull integrity, and overall vessel performance.

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