Substitute
, so that
![\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}\right]=-\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dz}+\dfrac1x\left(\dfrac1x\dfrac{\mathrm d^2y}{\mathrm dz^2}\right)=\dfrac1{x^2}\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dx%5E2%7D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cdfrac1x%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%5Cright%5D%3D-%5Cdfrac1%7Bx%5E2%7D%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%2B%5Cdfrac1x%5Cleft%28%5Cdfrac1x%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dz%5E2%7D%5Cright%29%3D%5Cdfrac1%7Bx%5E2%7D%5Cleft%28%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dz%5E2%7D-%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%5Cright%29)
Then the ODE becomes
which has the characteristic equation
with roots at
. This means the characteristic solution for
is
and in terms of
, this is
From the given initial conditions, we find
so the particular solution to the IVP is
An iterated integral is the outcome of taking integrals toward a function of more than one variable in such a way that part of the variables is treated as constants for each of the integrals.
From the given parameters, we are to write out five iterations for a triple integral;
where;
- the region is bounded by 0 ≤ z ≤ y, 0 ≤ y ≤ x², 0 ≤ x ≤ 8.
Thus, since x, y, z are functions of at least one variable, we can have the following iterations:
Learn more about iterated integrals here:
brainly.com/question/7009095
Answer:
y=3x+6
Step-by-step explanation:
The slope of the line is 3 and the equation will be y=3x+6
(Hard to put it in words for me so here is a picture)
Or you can use point-slope form (
) and insert the slope of -2 and the point (3,6) like this:
And simplify if needed.
Have a great day!
50 = 1/2 (20+ x)
100 = 20 + x
80 = x
answer
x = 80
