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June 30th, 2013, 08:26 AM  #1 
Newbie Joined: Sep 2012 Posts: 7 Thanks: 1  ODEs that are not quite simple for me.
Hello, I am not really great at arithmetic and just an average joe but this equations are killing me. I tried doing everything I know but nothing makes sense. Can anybody help me to find C in the following ODEs? 1. dy/dx = y/x 2. dy/dx = yx^2 Thank you for your response. 
June 30th, 2013, 09:01 AM  #2  
Senior Member Joined: Aug 2011 Posts: 334 Thanks: 8  Re: ODEs that are not quite simple for me. Quote:
What do you mean ? Can you explain what C is ?  
June 30th, 2013, 09:04 AM  #3 
Newbie Joined: Sep 2012 Posts: 7 Thanks: 1  Re: ODEs that are not quite simple for me.
we have to integrate both sides right? so in the first problem for example dy/dx = y/x dy/y=dx/x then to integrate: ln y = ln x + C the second problem is problematic because I get y= xy  x^3(1/3) + C 
June 30th, 2013, 09:15 AM  #4 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: ODEs that are not quite simple for me.
If no initial values are given, C is not expressible in terms of any number on R. However, one can do this : log y(x) = log(x) + C implies C = log y(1). And in y(x) = x * y(x)  x^3/3 + C, C = y(0). 
June 30th, 2013, 09:44 AM  #5 
Newbie Joined: Sep 2012 Posts: 7 Thanks: 1  Re: ODEs that are not quite simple for me.
You are supposed to let x = n. so any whone number can replace x. then we could solve for y. a friend of mine told me you are supposed to use integrating factors. on the 2nd problem. I got y=1+c. 
June 30th, 2013, 09:53 AM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: ODEs that are not quite simple for me. Quote:
Quote:
 
June 30th, 2013, 10:01 AM  #7  
Newbie Joined: Sep 2012 Posts: 7 Thanks: 1  Re: ODEs that are not quite simple for me. Quote:
 
June 30th, 2013, 10:04 AM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: ODEs that are not quite simple for me. Quote:
 
June 30th, 2013, 10:08 AM  #9 
Newbie Joined: Sep 2012 Posts: 7 Thanks: 1  Re: ODEs that are not quite simple for me.
oh... it is when y' + p(x)=q(x) then you multiply all sides by e^$p(x)dx

July 1st, 2013, 08:15 AM  #10 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: ODEs that are not quite simple for me.
dy/dx= y x^2 can be written as dy/dx y= x^2, a linear equation. An "integrating factor" for a this equation is a function, u(x), such that d(uy)/dx= u dy/dx+ uy. Using the product rule, u dy/dx+ u' y= udy/dx uy. That tells us u'= u which has solution u(x)= e^(x). That is, the equation is of the form e^x dy/dx e^x y= d(e^x y)/dx= x^2e^x. e^x y= . That last integral can be done by "integration by parts". 

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