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# solving differential equations

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1. Find both the first and second order differentials (y' and y") for the following functions:

2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.
2xy' = (y -x) (y + x)/ y

3. Solve the differential equation, y is a function of x
y'' - 6y' + 9y = x^2x

4. Solve the differential equation, y is a function of x
2y'' - 2y' + 5y = cos(x), y(0) = 1, y'(0) = 2

5. Solve the differential equation, y is a function of x
y'= 2x + 2xy, y(1)= 2

##### Solution Summary

It provides detailed explanations of different methods of solving differential equations, such as converting to exact ODE, undetermined coefficient method, and separation of variables.

##### Solution Preview

ET 7430

1. Find both the first and second order differentials (y' and y") for the following functions:

a.
Using the product rule and chain rule:

Then the second derivative is:

b.

Then the second derivative is:

c.
Apply the chain rule:

Apply the product rule and chain rule:

2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.

Rearrange the equation:


Then

And
is a function of x alone, then
is an integrating factor.
So multiplying 1/x2 to the equation (1):

(2)
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