What is the value of y?

Consider the differential equation:

Wewaii [24]

(a) Take the Laplace transform of both sides:

$2y''(t)+ty'(t)-2y(t)=14$

$\implies 2(s^2Y(s)-sy(0)-y'(0))-(Y(s)+sY'(s))-2Y(s)=\dfrac{14}s$

where the transform of $ty'(t)$ comes from

$L[ty'(t)]=-(L[y'(t)])'=-(sY(s)-y(0))'=-Y(s)-sY'(s)$

This yields the linear ODE,

$-sY'(s)+(2s^2-3)Y(s)=\dfrac{14}s$

Divides both sides by $-s$ :

$Y'(s)+\dfrac{3-2s^2}sY(s)=-\dfrac{14}{s^2}$

Find the integrating factor:

$\displaystyle\int\frac{3-2s^2}s\,\mathrm ds=3\ln|s|-s^2+C$

Multiply both sides of the ODE by $e^{3\ln|s|-s^2}=s^3e^{-s^2}$ :

$s^3e^{-s^2}Y'(s)+(3s^2-2s^4)e^{-s^2}Y(s)=-14se^{-s^2}$

The left side condenses into the derivative of a product:

$\left(s^3e^{-s^2}Y(s)\right)'=-14se^{-s^2}$

Integrate both sides and solve for $Y(s)$ :

$s^3e^{-s^2}Y(s)=7e^{-s^2}+C$

$Y(s)=\dfrac{7+Ce^{s^2}}{s^3}$

(b) Taking the inverse transform of both sides gives

$y(t)=\dfrac{7t^2}2+C\,L^{-1}\left[\dfrac{e^{s^2}}{s^3}\right]$

I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that $\frac{7t^2}2$ is one solution to the original ODE.

$y(t)=\dfrac{7t^2}2\implies y'(t)=7t\implies y''(t)=7$

Substitute these into the ODE to see everything checks out:

$2\cdot7+t\cdot7t-2\cdot\dfrac{7t^2}2=14$

5 0

3 years ago

Let segments AB, BC, and CA be tangent to circle G. If AB=52, BF=20, and CD=21, calculate the perimeter of triangle ABC. The ans

scoray [572]

Pls. see attachment.

The perimeter of the triangle is 146.

6 0

3 years ago

Solve the equation. Check your answer. 4|2x - 2|= 32

Eva8 [605]

Answer:

x

=

4

or

x

=

−

4

Explanation:

First divide throughout by

2

to get

x

2

=

16

Now take the square root on both sides to get

x

=

±

4

.

You may now substitute the values of

x

=

4

and

x

=

−

4

back into the original equation and check that it satisfies the equation which it does.

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers

2−1/2n=3n+16 n=3n+16

romanna [79]

Answer:

The answer is

n=5 3/5

Step-by-step explanation:

first collect the like terms:

2-1/2n=3n+16

then...

3n-1/2=16-2

==2 1/2n = 14

===n=5 3/5

3 0

3 years ago

I WILL MARK YOU BRAINLYEST if you answer all of my problems !!!!!!!!!!!

sweet-ann [11.9K]

1) you can buy 6 candy bars because $5.10 / $0.85 =
6.
To check: 6*0.85=5.10

2) you can find the answer by adding 3 to 12. 3+12=15 n=15 and it said n can be greater than or equal so
N(greater than or equal to sign) 15.
To check: 15-3=12
12=12
16-3>12
13>12

7) x-7=12
+7 +7
X=19
To check: 19-7=12
12=12

8) y/8=7
*8 *8
Y= 56
To check: 56/8=7
7=7

9) 10-4x=6x
+4x +4x
10=10x
/10 /10
X=1
To check: 10-4(1)=6(1)
10-4=6
6=6

Hope this helped!

3 0

3 years ago