February 9th, 2012, 02:34 AM  #1 
Member Joined: Jan 2012 Posts: 82 Thanks: 0  Integrating Factor Method
I am trying to use the Integrating Factor Method to find the general solution to the differential equation x^3 dy/dx = y+x^2e^4x After dividing through by x and rearranging I think I get dy/dy  1/x y = xe^4x Not sure this is correct Can anyone help? 
February 9th, 2012, 05:30 AM  #2 
Senior Member Joined: Dec 2011 Posts: 277 Thanks: 2  Re: Intergating Factor Method
To solve any first order linear ordinary differential equations by using integrating factor, we need to rewrite the given ODE into the standard form of . Notice that we need to have 1 as the coefficient of the term . Hence, from the given ODE , I find we need to divide the left and right sides of the given ODE by : 
February 9th, 2012, 10:09 AM  #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: Integrating Factor Method
Method for solving linear first order ODEs: a) Write the equation in the standard form b) Calculate the integrating factor by the formula: c) Multiply the equation in standard form by and, recalling that the lefthand side is just , obtain d) Integrate the last equation and solve for y by dividing by . The given ODE in standard form is: The integrating factor is then: and multiplying through, we have: The integral on the right cannot be expressed in elementary terms. 
February 9th, 2012, 06:50 PM  #4 
Senior Member Joined: Dec 2011 Posts: 277 Thanks: 2  Re: Intergating Factor Method
Argh...By rereading this post, to leave off at this step is what's killing me, but never mind, I'll blame my weary head in the middle of last night, . I really should write it in the form , i.e. Sorry and Mark's complete workout is great! 
February 10th, 2012, 06:09 AM  #5 
Member Joined: Jan 2012 Posts: 82 Thanks: 0  Re: Integrating Factor Method
Thanks for your help guys but I made a mistake, I should have written x dy/dx = y +x^2e^4x After dividing through by x we get dy/dx = y/x + x^2e^4x /x rearranging give dy/dx  y/x = x^2e^4x /x not sure about the rest. Can anyone help? 
February 10th, 2012, 06:20 AM  #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: Integrating Factor Method
Writing the ODE in standard linear form, we have: Now, we calculate the integrating factor: and multiplying through: 
February 11th, 2012, 01:20 AM  #7 
Member Joined: Jan 2012 Posts: 82 Thanks: 0  Re: Integrating Factor Method
Thank you so much for your help. I now need to find particular solution of the differential equation that satisfies the initial condition y(1)=0. So far I have For y(1)=0 0=1(1/4e^4*1 + C) Not sure about the algebra here to find out what C is. Can anyone help? 
February 11th, 2012, 01:26 AM  #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: Integrating Factor Method
You would simply have, upon solving for C, by subtracting from both sides: 
February 11th, 2012, 08:19 AM  #9 
Member Joined: Jan 2012 Posts: 82 Thanks: 0  Re: Integrating Factor Method
Would this then leave y = x (1/4^4x  1/4e^4) as the particular solution. 
February 11th, 2012, 08:45 AM  #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: Integrating Factor Method
No, it would be: 

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