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Zepler [3.9K]
3 years ago
14

Solve the differential equation by variation of parameters. y''+y=secx

Mathematics
1 answer:
suter [353]3 years ago
8 0

Answer:y=

Acosx+Bsinx +cosx ln(cosx)+x sinx

Step-by-step explanation:

given  equation y''+y=secx

auxiliary  equation

p^2+1=0\\p=\pm i

so CF is y=Acosx+Bsinx

now

y_1(x)=cosx \ \ \ \ y_2(x)=sinx\\{y_1}'(x)=-sinx \ \ \ \ {y_2}'(x)=cosx

using wronskian formula

W=\begin{vmatrix}cosx &sinx \\ -sinx & cosx\end{vmatrix}

          =cos^2x+sin^2x=1

now f(x)=secx

u=-\int \frac{f(x)y_2(x)}{W(x)}dx \ and \ v=\int \frac{f(x)y_1(x)}{W(x)}dx

u=-\int \frac{secx\times sinx}{1}dx \ and \ v=\int \frac{secx\times cosx}{1}dx

u=-\int tanx dx \ and \ v=\int {1}dx

u=ln(cosx) \ \ \ and \ \ v=x

now particular integrals are

PI=cosx ln(cosx)+x sinx

total solution

y= C.F+P.I

y=Acosx+Bsinx +cosx ln(cosx)+x sinx

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Answer:

D. (0, 3)

Step-by-step explanation:

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3 years ago
Taylor and Anya are friends who live 63 miles apart. Sometimes on a Saturday, they ride toward each other's houses on their bike
amid [387]

Answer:

Anya rode at 8.5 miles per hour.

Step-by-step explanation:

We have been given that Taylor and Anya live 63 miles apart. One day they left their houses at 8 am and met at 11 am riding toward each other. Taylor rode at 12.5 miles per hour.

We can see that Taylor and Anya met after 3 hours as 11-8=3. This means that both rode their cycles for 3 hours.

Let us find distance covered by Taylor.

\text{Distance}=\text{Speed*Time}

\text{Distance covered by Taylor}=12.5\frac{\text{Miles}}{\text{hour}}*3\text{ hours}}

\text{Distance covered by Taylor}=12.5*3\text{ Miles}

\text{Distance covered by Taylor}=37.5\text{ Miles}

Therefore, Taylor covered 37.5 miles in 3 hours.

Let us find distance covered by Anya in 3 hours by subtracting distance covered by Taylor from total distance of 63 miles.

\text{Distance covered by Anya in 3 hours}=(63-37.5)\text{ Miles}

\text{Distance covered by Anya in 3 hours}=25.5\text{ Miles}

Now let us find speed at which Anya rode bike.

\text{Speed}=\frac{\text{Distance}}{\text{Time}}

\text{Speed at which Anya rode the bike}=\frac{25.5\text{ Miles}}{3\text{ Hour}}

\text{Speed at which Anya rode the bike}=8.5\frac{\text{ Miles}}{\text{ Hour}}

Therefore, Anya rode the bike at the speed of 8.5 miles per hour.

6 0
3 years ago
Reduce to simplest form.<br> -5/8+(-8/5)=?
lara31 [8.8K]

Answer:

-2\frac{9}{40}

Step-by-step explanation:

-\frac{5}{8}+(-\frac{8}{5}) = -\frac{25}{40}-\frac{64}{40} (find common denominator)

=-\frac{89}{40}

=-2\frac{9}{40} (reduce to simplest form)

Hope this helps :)

4 0
2 years ago
Match each table showing a proportional relationship between x and y with its constant of proportionality.
Ann [662]

Answer:

Step-by-step explanation:

Just divide y by x and what ever you get is the answer

Hope this helps!

7 0
3 years ago
Read 2 more answers
What is a counterexample to this claim? Dividing a number by 2 always results in a smaller number.
In-s [12.5K]

Given:

Dividing a number by 2 always results in a smaller number.

To find:

The counterexample to the given claim.

Solution:

If 0 is divided by is divided by any non-zero real number a, then

\dfrac{0}{a}=0

Let us consider the unknown number be 0. Then dividing a number by 2, we get

\dfrac{0}{2}=0

Here, the result is not a smaller number because 0=0.

Therefore, the counterexample to the given claim is "Dividing 0 by 2".

7 0
2 years ago
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