(a) Take the Laplace transform of both sides:


where the transform of
comes from
![L[ty'(t)]=-(L[y'(t)])'=-(sY(s)-y(0))'=-Y(s)-sY'(s)](https://tex.z-dn.net/?f=L%5Bty%27%28t%29%5D%3D-%28L%5By%27%28t%29%5D%29%27%3D-%28sY%28s%29-y%280%29%29%27%3D-Y%28s%29-sY%27%28s%29)
This yields the linear ODE,

Divides both sides by
:

Find the integrating factor:

Multiply both sides of the ODE by
:

The left side condenses into the derivative of a product:

Integrate both sides and solve for
:


(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]](https://tex.z-dn.net/?f=y%28t%29%3D%5Cdfrac%7B7t%5E2%7D2%2BC%5C%2CL%5E%7B-1%7D%5Cleft%5B%5Cdfrac%7Be%5E%7Bs%5E2%7D%7D%7Bs%5E3%7D%5Cright%5D)
I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that
is one solution to the original ODE.

Substitute these into the ODE to see everything checks out:

Answer:
y=2x-5
Step-by-step explanation:
Slope is rise over run, or y/x
The points given were (0,-5) and (-1,-3)
Using y2-y1/x2-x1 we can find the slope.
-3-(-5)/-1-0 = 2/-1 = -2 = slope
Next, we have to find the y-intercept. The y-intercept is where the point on the line meets the y-axis.
This point is shown at -5.
Now, putting our information into y=mx+b (slope-intercept form)
y=2x-5
2 is our slope
-5 is our y intercept
Answer: V = (1/2 inches)^3 = 1^3/2^3 inches^3 = 1/8 Cubic inches.
Step-by-step explanation:
The formula of Volume for a cube is just like Hyperrectangle's V = width x length x height and since a Cube is just a Hyperrectangle with all sides equal, this specific case can be written as V = side^3
Answer:
Step-by-step explanation:
2
The coefficient is the number in front of the variable.
So, in this case, the coefficient of n^2 is 6 and of n is 2.
7x20 = 140 making that the total amount of decrease per week. Hope this helps!