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3 years ago

6

Help with number 11? Please

1 answer:

olasank [31]3 years ago

7 0

M(<BAC) = 180 - 70 - 65 = 45 degrees;
but, m(<A) < m(<B) < m(<C) => BC < AC < AB (3);

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Apply the method of undetermined coefficients to find a particular solution to the following system.wing system.

jarptica [38.1K]

$y''-y'+y=\sin x$

The corresponding homogeneous ODE has characteristic equation $r^2-r+1=0$ with roots at $r=\dfrac{1\pm\sqrt3}2$ , thus admitting the characteristic solution

$y_c=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x$

For the particular solution, assume one of the form

$y_p=a\sin x+b\cos x$

${y_p}'=a\cos x-b\sin x$

${y_p}''=-a\sin x-b\cos x$

Substituting into the ODE gives

$(-a\sin x-b\cos x)-(a\cos x-b\sin x)+(a\sin x+b\cos x)=\sin x$

$-b\cos x+a\sin x=\sin x$

$\implies a=1,b=0$

Then the general solution to this ODE is

$\boxed{y(x)=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x+\sin x}$

$y''-3y'+2y=e^x\sin x$

$\implies r^2-3r+2=(r-1)(r-2)=0\implies r=1,r=2$

$\implies y_c=C_1e^x+C_2e^{2x}$

Assume a solution of the form

$y_p=e^x(a\sin x+b\cos x)$

${y_p}'=e^x((a+b)\cos x+(a-b)\sin x)$

${y_p}''=2e^x(a\cos x-b\sin x)$

Substituting into the ODE gives

$2e^x(a\cos x-b\sin x)-3e^x((a+b)\cos x+(a-b)\sin x)+2e^x(a\sin x+b\cos x)=e^x\sin x$

$-e^x((a+b)\cos x+(a-b)\sin x)=e^x\sin x$

$\implies\begin{cases}-a-b=0\\-a+b=1\end{cases}\implies a=-\dfrac12,b=\dfrac12$

so the solution is

$\boxed{y(x)=C_1e^x+C_2e^{2x}-\dfrac{e^x}2(\sin x-\cos x)}$

$y''+y=x\cos(2x)$

$r^2+1=0\implies r=\pm i$

$\implies y_c=C_1\cos x+C_2\sin x$

Assume a solution of the form

$y_p=(ax+b)\cos(2x)+(cx+d)\sin(2x)$

${y_p}''=-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x)$

Substituting into the ODE gives

$(-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x))+((ax+b)\cos(2x)+(cx+d)\sin(2x))=x\cos(2x)$

$-(3ax+3b-4c)\cos(2x)-(3cx+3d+4a)\sin(2x)=x\cos(2x)$

$\implies\begin{cases}-3a=1\\-3b+4c=0\\-3c=0\\-4a-3d=0\end{cases}\implies a=-\dfrac13,b=c=0,d=\dfrac49$

so the solution is

$\boxed{y(x)=C_1\cos x+C_2\sin x-\dfrac13x\cos(2x)+\dfrac49\sin(2x)}$

7 0

3 years ago

-8x + 4 = 36 ggggggg

LekaFEV [45]

-8x+4=36
-4 -4
-8x=32
divde be -8 on both sides
x=-4

5 0

3 years ago

1) A museum charges different rates for adults and students. Adult

sveta [45]

Answer:

Step-by-step explanation:

6 0

3 years ago

3. Find an equation of a parabola with a vertex at the origin and directrix y = -3.5

Lina20 [59]

Answer:

$y=\frac{1}{14}x^2$

Step-by-step explanation:

we know that

The directrix of the parabola is perpendicular to the axis of symmetry of the parabola

In this problem the directrix is y=-3.5

so

The axis of symmetry is parallel to the y-axis

we have a vertical parabola

Also, the vertex is at the origin

That means-----> the parabola open upward

The equation of a vertical parabola can be written as

$x^2=4py$

where

p is the distance between the vertex and the directrix

In this problem

the distance between the vertex and the directrix is 3.5

$p=3.5$ ----> the value of p is positive because the parabola open upward

substitute

$x^2=4(3.5)y$

$x^2=14y$

isolate the variable y

$y=\frac{1}{14}x^2$

7 0

3 years ago

Read 2 more answers

During a clothing store’s Bargain Days, the regular price for T-shirts is discounted by $5. There is a state sales tax of 5%, an

ser-zykov [4K]

A) d = r - 5 where t is the full price
<span>b) p = ( r - 5 ) * 1.05 </span>
<span>c) p = (15.5-5)*1.05 = 10.5 * 1.05 = $ 11.02</span>

8 0

3 years ago

Read 2 more answers