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  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Please Solve the ODE
Wow. Six problems in an attachment requiring MSWord and no work shown. Quote:
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:
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:
We have the real root of multiplicity 2 and the complex roots and so we have:  
March 19th, 2012, 06:48 PM  #3  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Please solve the 6 ODEs... Quote:
Hence, the general solution is: Quote:
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:
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
ODEs are killing me!!!  zibb3r  Calculus  1  September 21st, 2013 01:35 PM 
ODEs that are not quite simple for me.  Execross02  Applied Math  9  July 1st, 2013 08:15 AM 
Solve the following ODEs...  FreaKariDunk  Calculus  12  February 22nd, 2012 08:59 PM 
System of odes, Problem  liakos  Applied Math  0  January 1st, 2011 09:52 AM 
Linearizing ODEs  acnash  Calculus  2  March 25th, 2010 04:19 PM 