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March 19th, 2012, 08:18 AM   #1
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Please solve the 6 ODEs...

Could you please solve the attached ODEs?
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March 19th, 2012, 11:17 AM   #2
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Re: Please Solve the ODE

Wow. Six problems in an attachment requiring MSWord and no work shown.

Quote:
1. Solve the ODE


Dividing through by x we obtain:



This is a first order homogeneous equation, so let:



With these substitutions, the ODE becomes:





This is a separable equation, so we have:



Integrating gives:





Substitute back for v:



Quote:
2. Solve the ODE


If we divide through by x and arrange the ODE as:



we have a Bernoulli equation. Dividing through by we obtain:



The substitution will give us the linear equation:



Divide through by x:











Substitute back for v:



where

We must observe that we lost the trivial solution in the above process.

Quote:
3. Find the complementary solution to the homogeneous ODE


The associated auxiliary equation is:





We have the real root of multiplicity 2 and the complex roots and so we have:

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March 19th, 2012, 06:48 PM   #3
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Re: Please solve the 6 ODEs...

Quote:
4. Find the general solution to the homogeneous ODE

where
The roots of the associated auxiliary equation are:



Hence, the general solution is:



Quote:
5. Use the method of undetermined coefficients to find the general solution of


First we determine the solution to the associated homogeneous equation, whose associate auxiliary equation has the roots:

hence:



Next, we determine the particular solution associated with the term

Since is a solution to the corresponding homogeneous equation, we assume the particular solution is of the form:







Substitution yields:







and so:



Now for the second particular solution associated with the term .

We may assume the particular solution is of the form:







Substitution yields:









Equating coefficients yields:





Solving this system yields:

and so:



Lastly, we determine the third particular solution associated with the term .

We may assume the particular solution is of the form:







Substitution yields:





Equating coefficients yields:



and so:



Then, the general solution is:



Quote:
6. Consider the ODE:



By using the substitution , show that the given differential equation can be converted into an ordinary differential equation with constant coefficients.

Solve this differential equation and hence obtain the solution of y as a function of x.
This is what's known as a Cauchy-Euler equation. The required substitution implies:

and it follows from the chain rule that:





Differentiating with respect to z:





Hence:



Substituting for and into the original ODE yields:





Now we have a homogeneous equation with constant coefficients, whose associated auxiliary equation has the roots:



and so the general solution is:



Substituting back for z we have:



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