(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:
You would end up with an equation of 8 - y/15.
Answer:
negative correlation
Step-by-step explanation:
because it starts from the y axis
I hope this helps you
radius=diameter/2
radius=1,5/2
radius=0,75
Area=pi. r^2
Area=3,14. (0,75)^2
Area=1,76
Answer:
4x³
Step-by-step explanation:
d/dx ln(x⁴ + 7) = 1/(x⁴ + 7) × _?
To obtain the missing expression, let us simplify d/dx ln(x⁴ + 7). This can be obtained as follow:
Let y = ln (x⁴ + 7)
Let u = (x⁴ + 7)
Therefore,
ln(x⁴ + 7) = ln u
Thus,
y = ln u
dy/du = 1/u
Next, we shall determine du/dx. This is illustrated below:
u = (x⁴ + 7)
du/dx = 4x³
Finally, we shall determine dy/dx of ln (x⁴ + 7) as follow:
dy/dx = dy/du × du/dx
dy/du = 1/u
du/dx = 4x³
dy/dx = 1/u × 4x³
But:
u = (x⁴ + 7)
Therefore,
dy/dx = 1/(x⁴ + 7) × 4x³
Thus, the missing expression is 4x³
