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July 22nd, 2013, 01:37 PM   #1
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First-Order, 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^2-xy) dy/dx = (1-xy)

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
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July 22nd, 2013, 03:42 PM   #2
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Re: First-Order, 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 back-substitute for and simplify.
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July 24th, 2013, 05:24 PM   #3
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Re: First-Order, 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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