### Exact equations

First determine if the equation is exact. If it is exact, find the general solution, or at least a relation that defines the solutions implicitly: [cos(x^2 + y) - 3xy^2]y' + 2x cos(x^2 + y) - y^3=0

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First determine if the equation is exact. If it is exact, find the general solution, or at least a relation that defines the solutions implicitly: [cos(x^2 + y) - 3xy^2]y' + 2x cos(x^2 + y) - y^3=0

Solve for either the general solution or relation. xy' - y = x(1 + e^(-y/x))

If the perimeter of a rectangle is 10 inches, and one side is one inch longer than the other, how long are the sides? Can you show me the steps to take to work out this and similar problems.

Diagonals in a Rectangle. In the case of a 2 X 2 rectangle, or a 3 X 5 rectangle, we can simply count. However, can we make a decision about a 100 X 167 or a 3600 X 288 rectangle? In general, given an N X K rectangle, how many grid squares are crossed by its diagonal?

Find the equation of a line L which passes through the point (1,4) and is perpendicular to the line: 6x + 3y=12.

The endpoints of the diameter of a circle are P=(-3,2) and Q=(5,-6) Find: (i) the center of the circle (ii) The radius of the circle (iii) the equation of the circle.

Logarithmic and Exponential Equations. See attached file for full problem description.

Solve for x: log2x=8 Write as a single logarithm: 5lnx-1n(x+1)

Log(x) means in this problem a log of base 3. ie, log(3)=1 sqrt(x) means the square root of x. ie, sqrt(25)=5 Solve the following equation for x: x^[log(9x)]=3sqrt(x)

Solve the integral using partial fraction decomposition. This example has a denominator that is the product of quadratics. Example 1) S (x2 + x +1)/[(x2 + 3x +1)(x2 +4x +2)] dx

2 - 13 divided by negative 76 + 15

D divided by ab(c to the 4th) plus c divided by a(b to the third) c

A 5000 gallon aquarium is maintained with a pumping system that circulates 100 gallons of water per minute through the tank. To treat a certain fish malady, a soluble antibiotic is introduced into the inflow system. Assume that the inflow concentration of medicine is 10te-t/50 oz/gal, where t is measured in minutes. The well-

I will use the ^ sign for the squared sign. Write the following algebraic expression in its simplest form: x^ + 2x + 3x^ + 2 + 4x + 7

Can you provide me with the basic steps for solving problems in algebra? Please provide an example (e.g., a problem solving for the variable x) using these steps.

1. Dan's father is 45. He is 15 years older than twice Dan's age. How old is Dan?

Suppose that n straight lines in the plane are positioned so that no two are parallel an no three pass throught the same point. Show that they divide the plane into 1/2(n^2 + n + 2) distinct regions.

A mapping %:A->B is called a constant map if there exists b.(b not) belonging to B such that %(a) = b. for all a belonging to A. Show that a mapping %:A->B is constant if and only if %$=% for all $:A->A

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I know the formula Rate x Time = Distance. However, I have a question where I am asked to find the time and not the distance? Can you give me some basic information about how to calculate the time using this formula or is there a different formula that I should be using? Thanks.

Karen can row a boat 10 kilometers per hour in still water. In a river where the current is 5 kilometers per hour, it takes her 4 hours longer to row a given distance upstream than to travel the same distance downstream. Find how long it takes her to row upstream, how long to row downstream, and how many kilometers she rows?

What if you want to calculate the number of miles a car travelling 30 mph goes in 120 minutes? How do I calculate this and with what formula?

What is the relationship between rate, time and distance? Say a car is traveling at 30 mph, how far will it go in 2 hours?

I will use parentheses for absolute value signs. I need help to solve these two equations. 1. 3(s) - 2 >7 and 2. 6 - (2 - p) < 4

Solve for the following scenario: The product of 2 consecutive positive odd numbers is 195. Find the numbers.

What must be added to each of the following expressions to obtain a perfect square? a) x^2 + 5x b) a^2 + 2ka c) c^2 - 4c

Find the equation in x whose roots are 2 and -3/4.

Solve the following quadratic equations by factorization: a) x^2 - 5x = 0 b) 6t^2 = t(t-4) c) a^2 + 9a = 0

1/X - X+1/8 = X - 1/4X . These are fractions.

Simplify y2 /X2 - 4y2 + y/2x + 4y When I have 2 after the y and the x it means squared. These are fractions, in case it is not clear.