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 July 22nd, 2013, 01:37 PM #1 Newbie   Joined: Jan 2013 Posts: 12 Thanks: 0 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
 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: First-Order, Exact and Linear Differential Equations 1.) This ODE is separable: $\frac{dy}{y-y^2}=\ln(x)\,dx$ Now, use partial fraction decomposition on the left, while on the right, integration by parts. 2.) Write the equation in the form: $(xy-1)\,dx+$$x^2-xy$$\,dy=0$ It is not exact, but an integrating factor of: $\mu(x)=e^{-\frac{1}{x}}=\frac{1}{x}$ will make the equation exact: $$$y-\frac{1}{x}$$\,dx+$$x-y$$\,dy=0$ 3.) Divide through by $x^2$ to obtain the first order homogeneous ODE: $\frac{dy}{dx}=-$$\frac{y}{x}$$^2-4$$\frac{y}{x}$$$ Now, using the substitution: $v=\frac{y}{x}\,\therefore\,\frac{dy}{dx}=x\frac{dv }{dx}+v$ we now have: $x\frac{dv}{dx}+v=-v^2-4v$ $x\frac{dv}{dx}=-v^2-5v$ This ODE is separable: $\frac{dv}{v^2+5v}=-\frac{dx}{x}$ Now, use partial fraction decomposition on the left, while on the right, obtain a logarithmic function. Then back-substitute for $v$ and simplify.
 July 24th, 2013, 05:24 PM #3 Newbie   Joined: Jan 2013 Posts: 12 Thanks: 0 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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# how to solve linear differential equations of first order

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