If U = x y, find dU/dt if x* + y = t and x + y° = f
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
6.29) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
![Implicit functions and jacobians
6.29.
If U = x y, find dU/dt if
x' + y = t and
(1)
x² + y³ = ?
Equations (1) and (2) define x and y as (implicit) functions of t. Then differentiating with respect to t, we
have
5x* (dx/dt) + dy/t = 1
(3)
2x (dx/dt) + 3y²(dy/dt) = 2t
(4)
Solving Equations (3) and (4) simultaneously for dx/dt and dy/dt,
1
1
5x*
1
2t 3y
5x
2х Зу
Зу? - 21
15х*у? — 2х
dx
dy
2x
2t
10х*t - 2х
dt
1
dt
5x*
1
15х*у? - 2х
2х Зу
10x*t - 2х
dU
Then
dt
JU dx
ax dt
JU dy
= (3x²y)|
ду dt
Зу? - 2t
15х*у? - 2х
15x*y – 2x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad530511-9758-42d9-9994-894bc4eea3f3%2Ff8f4c3c4-2013-4acf-9396-b0eeb261f3f3%2Fkx7gwqj_processed.png&w=3840&q=75)
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Given
Then
Differentiate equation (1) and equation (2) with respect to , we get
From equation (3) and equation (4) , we will find value of and
We will use Cramer's rule to determine this
Equation (3) and equation (4) in the matrix form is given by where
Then
Then is formed by replacing the first column of matrix with the matrix and is formed by replacing the second column of matrix with the matrix
Thus
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