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 March 19th, 2012, 08:18 AM #1 Newbie   Joined: Mar 2012 Posts: 4 Thanks: 0 Please solve the 6 ODEs... Could you please solve the attached ODEs? March 19th, 2012, 11:17 AM   #2
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Math Focus: Calculus/ODEs
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: March 19th, 2012, 06:48 PM   #3
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From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
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Math Focus: Calculus/ODEs
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: Tags odes, solve Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zibb3r Calculus 1 September 21st, 2013 01:35 PM Execross02 Applied Math 9 July 1st, 2013 08:15 AM FreaKariDunk Calculus 12 February 22nd, 2012 08:59 PM liakos Applied Math 0 January 1st, 2011 09:52 AM acnash Calculus 2 March 25th, 2010 04:19 PM

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