solving differential equations

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)
Whose solution is obtained as follows:

Then

Where is the coefficient of dx in equation (2)
So

Therefore,

And

Which implies that

And the general solution is

3. Solve the differential equation, y is a function of x

The characteristic equation of the homogeneous ODE is

It has double roots
Then the general solution to the homogeneous ODE is

Now, since the right side function is an exponential and we know that exponentials never just appear or disappear in the differentiation process it seems that a likely form of the particular solution would be

Now, all that we need to do is do a couple of derivatives, plug this into the differential equation and see if we can determine what A needs to be.

Then

So a particular solution to the differential equation is then,

Thus, the general solution to the differential equation is

4. Solve the differential equation, y is a function of x

For this undetermined coefficient type of ODE, since the right-hand function is , we guess that the particular solution is

Now differentiate it:

Substituting back gives:

Compare the coefficients, we have

Solve for A and B:

So the particular solution is

Now find the general solution to the homogenous equation
The roots of its characteristic equation are:

Then general solution to the homogenous ODE is

Therefore, the general solution to the given ODE is:

Now find the constants c1 and c2 using the initial values:

Differentiate y(x):

Then

Thus the solution of this IVP is:

5. Solve the differential equation, y is a function of x

So this DE can be solved by separation of variables.
Integrate on both sides:

Then

Substitute 1 for x to find constant A:

So the solution is

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