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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 March 22nd, 2011, 04:33 PM #11 Senior Member   Joined: Feb 2011 Posts: 150 Thanks: 0 Re: 2nd Order Differential Equation the (x^2) should be in there.. That comes from the y=c1 x^2 e^x +c2 e^-x. I see Ok great thanks. On another subject, if a problem d^2 x/dt^2 -4 dx/dt = 7-3e^4t d^2 x/dt^2 -4 dx/dt =0 , Auxillary equation, if it is of the form m^2 -4m =0, What would it be for m=? , as I realise if the auxillary equation is of the form m^2 -4 then, m= +or- 2. I should know this ! March 22nd, 2011, 05:41 PM #12 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: 2nd Order Differential Equation We begin with: Let's go another route with this one. Multiply through by : Now, notice the left-side is the derivative of a product: Now, multiply through by Note: the 2nd term in the integrand on the right requires integration by parts. Since is a constant, we may represent the constant simply as . Now, had we worked this equation by the method of undetermined coefficients, we would first find: Next, we would assume a particular solution of the form: (we need the t as a factor on both terms so that neither term is a form of ) Now, substituting these into the original ODE gives: Equating coefficients: Thus, we have: So, putting it together, we have: Since the parameters are constants, we may write this in the form: March 24th, 2011, 07:08 AM #13 Senior Member   Joined: Feb 2011 Posts: 150 Thanks: 0 Re: 2nd Order Differential Equation x dy/dx -2y = x^4 e^x Divide by x^3 x/x^3 dy/dx -2y/x^3 = x^4 e^x /x^3 x^-2 dy/dx -2x^-3 y = x e^x d/dx y/x^2 = x e^x. Exact form Integrate wrt x d(y/x^2) =x e^x dx y = x^2 (e^x (x-1) + c ) Can you point out my errors, I'm sure there will be some ! Thanks March 24th, 2011, 09:51 AM #14 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: 2nd Order Differential Equation No errors, looks good!  March 26th, 2011, 02:01 AM #15 Senior Member   Joined: Feb 2011 Posts: 150 Thanks: 0 Re: 2nd Order Differential Equation Ok so, say I had an equation similar to the 2nd order equation you helped me with before apart from the right side being slightly different. d^2 x/dt^2 -4 dx/dt = 7-3te^4t Very similar to previous one however there is '3t' in there now. My question is what form would the complimentry function xh(t) be and what could I try the particular solution xp(t) as. Could the right side be re-written or am I just looking to much into it? Jake March 26th, 2011, 04:58 AM #16 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: 2nd Order Differential Equation Since the left side is the same, the complimentary function would be the same: For the particular solution, we could use: Differentiating, substituting into the ODE, and solving for the undetermined coefficients gives: Adding this to the complimentary function gives: March 26th, 2011, 07:14 AM #17 Senior Member   Joined: Feb 2011 Posts: 150 Thanks: 0 Re: 2nd Order Differential Equation Awesome , but backing up bit I need to get the 1st and 2nd derivative right I'n my mind, I got x'p(t)= A(2+B)4e^4t + C and x''p(t)=8e^4t(A(B+2t)+2A(B+2t)) is this correct? March 26th, 2011, 11:44 AM #18 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: 2nd Order Differential Equation We have: Using the product rule, we get (after simplification): March 26th, 2011, 12:47 PM #19 Senior Member   Joined: Feb 2011 Posts: 150 Thanks: 0 Re: 2nd Order Differential Equation Ok I was way out. after subbing into the original equation, what is the end equation after simplfying.... Its a long one. So I can compare coefficients myself  March 26th, 2011, 12:50 PM #20 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: 2nd Order Differential Equation After subbing into the original ODE and simplifying, I got: Tags 2nd, differential, equation, order Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post charmi Differential Equations 0 December 21st, 2013 03:08 AM Survivornic Differential Equations 2 October 29th, 2012 12:17 PM jo2jo Differential Equations 3 March 16th, 2012 06:50 AM Foolish Differential Equations 0 October 13th, 2010 12:46 PM Seng Peter Thao Differential Equations 0 June 30th, 2007 10:55 AM

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