Question

1. For each of the following pairs of equations, find the x and y values at the intersection

a) 36=xy and (y)/(x)=(1)/(4)

b) 4=x^((1)/(2))y^((1)/(2)) and y/x=1/4

c) a=xy and y/x=b/c

d) 48=4x+6y and y/x=2/3

Answer #1

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

Find the absolute maximum and minimum values of f on
the set D.
f(x, y) =
4x + 6y −
x2 − y2 +
2,
D = {(x,
y) | 0 ≤ x ≤ 4, 0 ≤
y ≤ 5}

Find the absolute maximum and minimum values of f on
the set D.
f(x, y) =
4x + 6y −
x2 − y2 +
8,
D = {(x,
y) | 0 ≤ x ≤ 4, 0 ≤
y ≤ 5}
Find the absolute maximum and minimum values of f on
the set D.
f(x, y) = 2x3 + y4 +
2, D = {(x, y) | x2 +
y2 ≤ 1}

Find the absolute maximum and minimum values of f on
the set D.
f(x, y) =
4x + 6y −
x2 − y2 +
3,
D = {(x,
y) | 0 ≤ x ≤ 4, 0 ≤
y ≤ 5}
absolute maximum value
absolute minimum value

Find the general solution of the following differential
equations. Primes denote derivates with respect to x.
1) x(x+3y)y'= y(x-3y)
2) 3xy^2y'= 21x^3+3y^3
3) x^2y'= xy+10y^2
4) x(4x+3y)y'+ y(12x+3y)= 0
5)2xyy' = 2y^2 + 7xsqrt(9x^2+y^2)

Find a particular solution to each of the following differential
equations with the given initial condition:
A) dy/dx=ysinx/1+y^2, y(0)=1
B)dy/dx=xy ln x, y(1)=3
C)xy(1+x^2) dy/dx=(1+y^2), y(1)=0

Suppose you are given the following (x, y)
data pairs.
x
1
2
6
y
4
3
9
Find the least-squares equation for these data (rounded to three
digits after the decimal).
ŷ = + x
(b) Now suppose you are given these (x, y) data
pairs.
x
4
3
9
y
1
2
6
Find the least-squares equation for these data (rounded to three
digits after the decimal).
ŷ = + x
(c) In the data for parts (a) and (b), did we...

Given each of the equations below, find dy/dx:
(a)x^2y=x^2+y^2
(b) (x+1)/x= 1−xy
(c) siny=x^2−1

Find the absolute maximum and minimum values of f on
the set D.
f(x, y) =
4x + 6y −
x2 − y2 +
5,
D = {(x,
y) | 0 ≤ x ≤ 4, 0 ≤
y ≤ 5}
absolute maximum value
absolute minimum value

Consider the following system of differential equations dx/dt =
(x^2 + 2x + 1)(x^2 − 4x + 4) dy/dt = xy − 1
Which of the following is not an equilibrium point of the above
system? (A) (3, 1/3 ) (B) (−1, −1) (C) (1, 1) (D) (1, 3)

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