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DerKrebs [107]
3 years ago
13

Find two power series solutions of the given differential equation about the ordinary point x = 0. compare the series solutions

with the solutions of the differential equation obtained using the method of section 4.3. try to explain any differences between the two forms of the solution. y'' − y' = 0
Mathematics
1 answer:
monitta3 years ago
4 0
I don't know what method is referred to in "section 4.3", but I'll suppose it's reduction of order and use that to find the exact solution. Take z=y', so that z'=y'' and we're left with the ODE linear in z:

y''-y'=0\implies z'-z=0\implies z=C_1e^x\implies y=C_1e^x+C_2

Now suppose y has a power series expansion

y=\displaystyle\sum_{n\ge0}a_nx^n
\implies y'=\displaystyle\sum_{n\ge1}na_nx^{n-1}
\implies y''=\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}

Then the ODE can be written as

\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge1}na_nx^{n-1}=0

\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge2}(n-1)a_{n-1}x^{n-2}=0

\displaystyle\sum_{n\ge2}\bigg[n(n-1)a_n-(n-1)a_{n-1}\bigg]x^{n-2}=0

All the coefficients of the series vanish, and setting x=0 in the power series forms for y and y' tell us that y(0)=a_0 and y'(0)=a_1, so we get the recurrence

\begin{cases}a_0=a_0\\\\a_1=a_1\\\\a_n=\dfrac{a_{n-1}}n&\text{for }n\ge2\end{cases}

We can solve explicitly for a_n quite easily:

a_n=\dfrac{a_{n-1}}n\implies a_{n-1}=\dfrac{a_{n-2}}{n-1}\implies a_n=\dfrac{a_{n-2}}{n(n-1)}

and so on. Continuing in this way we end up with

a_n=\dfrac{a_1}{n!}

so that the solution to the ODE is

y(x)=\displaystyle\sum_{n\ge0}\dfrac{a_1}{n!}x^n=a_1+a_1x+\dfrac{a_1}2x^2+\cdots=a_1e^x

We also require the solution to satisfy y(0)=a_0, which we can do easily by adding and subtracting a constant as needed:

y(x)=a_0-a_1+a_1+\displaystyle\sum_{n\ge1}\dfrac{a_1}{n!}x^n=\underbrace{a_0-a_1}_{C_2}+\underbrace{a_1}_{C_1}\displaystyle\sum_{n\ge0}\frac{x^n}{n!}
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There are 2.54 centimeters in 1 inch. There are 100 centimeters in 1 meter.
Alika [10]

Answer:

5 meters

Step-by-step explanation:

If in 1 inch have 2.54 centimeters, in 197 inches will have:

197×2.54 = 500.38 centimeters

If 100 centimeters are 1 meter, 500 centimeters will have 5 meters, so 197 inches have 5 meters

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3 years ago
Setting 7/3 equal to which ratio would result in a valid proportion? 9/49 18/42 42/18 49/9<br>​
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2 years ago
The vertices of a triangle are (-1,3) (6,3) and (-1,-4)
Travka [436]

Answer:

24.5 unit²

Step-by-step explanation:

Area of ∆

= ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |

= ½ | (-1)(3 -(-4)) + 6(-4 -3) + (-1)(3 - 3) |

= ½ | -7 - 42 |

= ½ | - 49 |

= ½ (49)

= 24.5 unit²

<u>Method 2:</u>

Let the vertices are A, B and C. Using distance formula:

AB = √(-1-6)² + (3-3)² = 7

BC = √(-6-1)² + (-4-3)² = 7√2

AC = √(-1-(-1))² + (4-(-3))² = 7

Semi-perimeter = (7+7+7√2)/2

= (14+7√2)/2

Using herons formula:

Area = √s(s - a)(s - b)(s - c)

here,

s = semi-perimeter = (14 + 7√2)/2

s - a = S - AB = (14+7√2)/2 - 7 = (7 + √2)/2

s - b = (14+7√2)/2 - 7√2 = (14 - 7√2)/2

s - c = (14+7√2)/2 - 7 = (7 + √2)/2

Hence, on solving for area using herons formula, area = 49/2 = 24.5 unit²

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2 years ago
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20/-4 = -5

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What is another word for magnitude?
Marianna [84]
The size of something. Typically something that is very large.
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