Please see the attached file for the complete solution.
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Substitution Methods and Exact Equations

Homogeneous Equations:
Dy/dx = F(y/x)

v = y/x, y = vx, dy/dx = v + x(dv/dx)

x(dv/dx) = F(v) - v

Bernoulli Equations:

dv/dx + (1-n) P(x)v = (1-n) Q(x)

Criterion for Exactness:

F(x,y) = ∫ M(x, y) dx + g(y)

Verify that the given differential equation is exact; then solve it.

31. (2x + 3y) dx + (3x + 2y) dy = 0

Answer: x2 + 3xy +y2 = C

Solution. This differential equation is in the form of P(x,y)dx+Q(x,y)dy=0, where P(x,y)=2x+3y and Q(x,y)=3x+2y.

Note that . So, there exists a function Z(x,y) such that dZ=P(x,y)dx+Q(x,y)dy. Hence, the general solution to P(x,y)dx+Q(x,y)dy=0 is

Z(x,y)=C, where C is a constant.

Now the question becomes to find such function Z(x,y). We can do it as follows.

As , we can integrate it with respect to x to ...

Solution Summary

Differential equatiosn are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

1. Please see the attached file for the fully formatted problems.
a) Use separation of variables to solve
b) Solve the following exactdifferential equation
c) By means of substitution y=vx solve the differential equation
d) By means of the substitution for approipriate values of and , solve the d

Differential Equation (IX): Formation of DifferentialEquations by Elimination
Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

A) Solve the following differential equation by as many different methods as you can.
(See attachment for equation)
b) There is a type of differential equation which will always be solvable by two different methods. What type of differential equation is it and which other method can always be used to solve it?
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1. Solve the following system of equations using the Substitution method:
5X-7Y= - 18
4X+3Y= 20
2. Solve the following system of equations using the elimination method:
2X-3Y=12
-2X+3Y=12
Solution: Computation of the Equations
2X-3Y =12
-2X+3Y =12
3. A total of $80,000 is invested in two funds paying 2.3% and 3.1

I am asking for the step-by-step workings for all of the attached problems.
** Please see the attached file for complete problem description **
1st problems. Please find the general solution of:
(1) dy/dx = y/sin(y) - x
(2) dy/dx = y + cos(x)y^2010
In the process of finding the solutions for the problems make use of both

4x+y=12 (1)
x-y=8(2)
4x+y=12
x-y=8
5x/5 = 20/5
X=4
4(4) =y - 12
16+y = 12
Y =4
I think this would be an example of elimination method for solving a system of equations however I am unsure how it would transfer to substitution method thus I am needing assistance. I am needing this to be illustrated if you wi

Please help with the following problems.
There are three methods to solving Linear Systems with two Equations. They are the Graph method, the Elimination method, and the Substitution method. When would you use each method? What makes each method better than the other methods?