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March 11th, 2011, 10:00 AM  #1 
Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0  Linear differential equations
Problem: Use the integrating factor method to solve the following linear differential equation x dy/dx  y = x I used the same method as I did last night with the other equations and found the integrating factor to be ?(x)=e^?1/x dx = e^ln(x) = x (or in this case would it be just (x) ? ) Then Multiplying the standard form equation by (x) , x(dy/dx)y= x Integrating to get y=1/2 x + A / x Im a bit unsure on it 
March 11th, 2011, 10:12 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: Linear differential equations Now try it. 
March 11th, 2011, 10:26 AM  #3 
Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0  Re: Linear differential equations
y=  (ln(x) +A / x)

March 11th, 2011, 10:36 AM  #4 
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: Linear differential equations
We have in standard form: Multiplying through by gives: Integrate w.r.t. x: 
March 11th, 2011, 11:38 AM  #5 
Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0  Re: Linear differential equations
oh yes, i made the mistake on multiplying by the integrating factor. my next equation cos(t) dx/dt + x sin(t) = cos^2 (t) Re write in Std.form, divide by cos(t) dx/dt + x (sin(t)/cos(t)) = cos(t) Obtain Integrating factor, ?(x)=e^?sin(t)/cos(t) = e^ln(cos(t)) = cos(t) or is it 1/cos(t) 
March 11th, 2011, 12:37 PM  #6 
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: Linear differential equations
It is , since by the properties of logarithms, we have . This transforms your equation in standard form to: Then you have: 
March 11th, 2011, 01:09 PM  #7 
Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0  Re: Linear differential equations
oh ok, yes i thought the laws of logs would be involved. so then i integrate d/dx (x sect)=1 ? 
March 11th, 2011, 01:19 PM  #8 
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: Linear differential equations
You integrate: w.r.t. t (Notice the slight difference in what I wrote and what you wrote. ) 
March 11th, 2011, 01:32 PM  #9 
Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0  Re: Linear differential equations
oh yes it was suposed to say t. x sec(t) = x + C , do i divide by sec(t) 
March 11th, 2011, 02:21 PM  #10 
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: Linear differential equations
Or multiply through by cos(t), same thing. Although your right side should be t + C (recall we integrated with respect to t). 

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