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July 22nd, 2013, 01:37 PM  #1 
Newbie Joined: Jan 2013 Posts: 12 Thanks: 0  FirstOrder, Exact and Linear Differential Equations
Hi all, I'm having a hard time differentiating how to solve these type of equations, and I was hoping to post 3 examples and get some help: 1) Solve dy/dx + y ln(x^y) = y lnx, x>0 2) Solve (x^2xy) dy/dx = (1xy) 3) Solve x^2y' + y^2 + 4xy = 0, x does not equal 0 I'm having a mental block with these type of problems, if someone could lend a hand and explain the process that would be greatly appreciated. Brad 
July 22nd, 2013, 03:42 PM  #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: FirstOrder, Exact and Linear Differential Equations
1.) This ODE is separable: Now, use partial fraction decomposition on the left, while on the right, integration by parts. 2.) Write the equation in the form: It is not exact, but an integrating factor of: will make the equation exact: 3.) Divide through by to obtain the first order homogeneous ODE: Now, using the substitution: we now have: This ODE is separable: Now, use partial fraction decomposition on the left, while on the right, obtain a logarithmic function. Then backsubstitute for and simplify. 
July 24th, 2013, 05:24 PM  #3 
Newbie Joined: Jan 2013 Posts: 12 Thanks: 0  Re: FirstOrder, Exact and Linear Differential Equations
Hi Mark, Thank you for your help. I'm going to work through the solutions, and post them in a day or so, just to make sure I complete them properly. I'm teaching myself this stuff through distance education, and am having a hard time, so I appreciate your help ! 

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