# 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

https://brainmass.com/math/calculus-and-analysis/solving-differential-equations-433751

#### Solution Preview

ET 7430

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

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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:

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Then

And

is a function of x alone, then

is an integrating factor.

So multiplying 1/x2 to the equation (1):

(2)

Whose ...

#### 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.