
For the ODE to be exact, we require that

, which we'll verify is not the case here.


So we distribute an integrating factor

across both sides of the ODE to get

Now for the ODE to be exact, we require

, which in turn means

Suppose

. Then substituting everything into the PDE above, we have




so that our integrating factor is

. Our ODE is now

Renaming

and

to our current coefficients, we end up with partial derivatives


as desired, so our new ODE is indeed exact.
Next, we're looking for a solution of the form

. By the chain rule, we have

Differentiating with respect to

yields


Thus the solution to the ODE is
Answer:
The graph shifts 7 units left and 2 units up.
Step-by-step explanation:
In transformations, when a number is added or subtracted to "x" directly, that is the left/right shift.
When a number is added or subtracted outside of the "rest" of the equation, that is the up/down shift.
I learned transformations with these variables:

"a" - vertical stretch (a > 1) or compression (0 < a < 1)
"k" - horizontal stretch (0 < k < 1) or compression (k > 1)
"c" - translate left (+) or right (-)
"d" - translation up (+) or down (-)
The untranslated graph is centred around the origin f(x) = 1/x
f(x) begins 3 units right and 1 unit up of the untranslated graph.
g(x) is 4 units left and 3 units up of the untranslated graph.
y
g |
| f
-----------|------------ x
There is a 7 units translation left and 2 units translation up.
Answer: A: 6
B: 4
C: 2
Step-by-step explanation:
The value of x when y = 14 is 8.4
<h3 />
<h3>How to solve variation ?</h3>
y, varies directly as x. Therefore,
y ∝ x
Therefore,
y = kx
where
- k =constant of proportionality
Therefore,
10 = 6k
k = 10 / 6
k = 5 / 3
Hence, when y = 14
y = kx
14 = 5 / 3 x
cross multiply
42 = 5x
x = 42 / 5
x = 8.4
learn more on variation here: brainly.com/question/20707320
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