2nd order Definition and 494 Threads

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  1. J

    How can I convert a 2nd order ODE to a 1st order ODE?

    Hi, have this strange 2nd order ODE in one of my tutorials that I am struggling to start. I am not used to dealing with derivatives of both x and y as well as a function involving t. I was wondering if anyone may be able to point me to the starting line. I am trying to convert them into 1st...
  2. P

    How to Solve a Second Order Differential Equation?

    what's the method to solve a diff. equation as follows: d^2(psi)/d(x^2) - (k^2)*(psi) = 0
  3. G

    How Can You Solve an Integral with a 2nd Order Pole on the Real Axis?

    I'm working on a complex analysis problem from Arfken. Integrate[ (cos(b*x)-cos(a*x))/(x^2), {x, -Infinity, Infinity}] and show that it is equal to Pi*(a-b) Attempt: I first look at a related problem (or one that I think is related). => f(x) = sin(x)/x => f(z) = sin(z)/z...
  4. D

    Solve 2nd Order ODE Mirror for Parallel Reflection from Origin

    Homework Statement A curved mirror of equation y=y(x) has that property that whenever a ray of light emanates from the origin it reflects parallel to the x-axis. Find the equation of the mirror Don't even know how to get started on this, Don't need a solution just need some starting hints...
  5. A

    2nd order DE violating theroem 2?

    Homework Statement y1 = x2 and y2 = x3 are two different solutions of x2y'' - 4xy' + 6y = 0, both satisfying the initial conditions y(0) = 0 = y'(0). Explain why these facts don't contradict Theorem 2 (with respect to the guaranteed uniqueness). Homework Equations The Attempt at a...
  6. A

    Solve IVP 2nd order differential equation

    Homework Statement Find the particular solution of the linear, homogeneous, 2nd order differential equation: y'' - 2y' + 2y = 0, given the solutions y1 = (e^x)*(cos x), y2 = (e^x)*(sin x), y(0) = 0, y'(0) = 5. Homework Equations The Attempt at a Solution How do I begin? I'd really...
  7. A

    Verify y1 solves 2nd order differential equation

    Homework Statement Verify that y1 = (e^x) * (cos x) solves the linear, homogeneous, 2nd order differential equation y'' - 2y' + 2y = 0 Homework Equations The Attempt at a Solution So I used the product rule but just kept going in circles with the cos x. I remember there's a way to...
  8. H

    Mathematica Mathematica: 2nd order PDE, variable coefficients

    a\text{''}[t]+B[t]*a'[t]-A[t]*a[t]==0 a[0] = 10^-9 a'[0] = 0 a[t] = ? The coefficients A and B are variable over time. I HAVE solved (experimental and theoretical values) for the values of A and B over the time interval of interest! I attempted to solve for a[t] using NDSolve as one...
  9. H

    Mathematica: 2nd order PDE variable coefficients

    a\text{''}[t]+B[t]*a'[t]-A[t]*a[t]==0 a[0] = 10^-9 a'[0] = 0 a[t] = ? The coefficients A and B are variable over time. I HAVE solved (experimental and theoretical values) for the values of A and B over the time interval of interest! I attempted to solve for a[t] using NDSolve as one...
  10. Saladsamurai

    Solving a 2nd Order ODE: P^2 - 4xP +6y = 0

    Homework Statement \left(\frac{dy}{dx}\right)^2 - 4x\frac{dy}{dx} + 6y = 0 Homework Equations A common approach we have used for similar problems has been to let P = dy/dxThe Attempt at a Solution Doing so we have: P^2 - 4xP +6y = 0 \Rightarrow 6y = 4P(x - P) Differentiating gives: 6P...
  11. L

    Finding r for 2nd Order ODE Solutions: e^rt and te^rt | Homework Help

    Homework Statement Find a value of the constant r such that both e^rt and te^rt are solutions to the ODE ay''+by'+cy=0 Homework Equations The Attempt at a Solution can anyone guide me with this question please. I am not sure where to start. I know that e^rt is always a solution...
  12. L

    2nd order homogeneous equations complex root

    Homework Statement y'' -2y' +5y =0 , y(0)=1, y'(0)=1 you get a complex root conjugate. Homework Equations y=e^(rt) y'=re^(rt) y''=r^2 * e^(rt) The Attempt at a Solution I have in my notes sin(omega*t)e^(sigma *t), cos(omega *t)e^(sigma). I don't think i took down the notes...
  13. A

    Runge-Kutta 4 with 2nd Order: Solving Variables (x,y,Q)

    i would like to programing in MATLAB with runge kutta 4 with 2nd order the variables (x with y and Q) where x distance , y hight of water and Q discharge from the following tow equation dydx = (s0-(((Q0^2*n^2)/(B^2.*y^(10/3))*(1+2.*y/B)^(4/3))+dQdx))/(1-Q0^2/(g*B^2*y.^3)); dQdx =...
  14. P

    How can I solve the 2nd order PDE for \beta^{(0)} in geochemical thermodynamics?

    Hi, I've spent days trying to solve some equations in a paper (referenced below) that describes it as a "straightforward, albeit lengthy integration," but I can't work out the "straightforward" bit. The notation is also odd, which doesn't seem to help my problem. Perhaps someone could help...
  15. F

    2nd order nonlinear differential equation

    Hello everybody, could you please direct me how to solve this nonlinear differential equation analytically, so by mathematica or matlab? I really need to solve it for my research project, so please help me du/dx=d/dx[a*u^(-1/2)*du/dx]-n*u^(3/2)*(u-c)/b boundry conditions are: u(0)=b+c...
  16. B

    Solving 2nd Order Homogeneous Equations with Non-Constant Coefficients

    Hi Guys, I know how to find the solution to a 2nd order homogeneous with constant coefficients but how do you solve one with a non constant ie x^2y''+2xy' ... etc = 0 Is there a general solution formula for these types of problems? My book seems to jump from 2nd order...
  17. Rasalhague

    Is to 2nd order always related to Taylor series?

    What does "to 2nd order" mean? http://www.mathpages.com/rr/s2-11/2-11.htm I thought "to second order" meant an approximation consisting of the initial terms of the Taylor series up to and including the second derivative term. For example, it seems to be used that way in the Wikipedia...
  18. N

    Solving a Nonhomogeneous 2nd Order ODE with Initial Conditions

    Homework Statement This was an example the teacher gave us to do in class. Unfortunately, I was a little behind on the notes so was not able to copy down the steps to solving this one. I did, however, get an answer problem: y'' + y = e^x + x^3, initial conditions y(0)=2, y'(0)=0...
  19. G

    Solving a Second Order Differential Equation with Complex Roots

    I have put down the whole question and my answers which i would appreciate if someone could comment: Part 1.1 Find the general solution of the equation (d^2 z)/(dx^2 ) - 4 * dz/dx + 13z = 0 Put into form: a * m^2 + b * m + c ∴ m^2 - 4m + 13 = 0 a = 1, b = -4, c = 13 Then: m_(1,2 =...
  20. C

    4th order Runge Kutta method for 2nd order ODE

    Hello, i have a bit of a problem with uderestanding how exactly we use RK4 method for solving 2nd order ODE. And last conversation with my proffesor only added up to my confiusion. Further more i couldn't find any example dealing with this problem if any1 could provide link explaining this...
  21. P

    Relating 2nd order partial derivatives in a coordinate transformation.

    Homework Statement Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind. Suppose I have two functions \phi(t) and \chi(t) and the potential V which...
  22. A

    2nd order differential equation

    Homework Statement y^{2}\frac{d^{2}y}{dx^2} + ay = b(cx-d) Find y as a function of x,a,b,c & d (a,b,c & d are all constant(!)) Homework Equations - The Attempt at a Solution Not a clue, this is actually how far I got with my own take on an orbital mechanics problem I made up...
  23. X

    Impulse Response for a 2nd Order Diffy Q

    y'' + 6y' + 4y = x(t) I found the eigenvalues to be -.764 and -5.24, no problems here. Next, since the order of the response is greater than the order of the excitation, I assumed the unit response took the form: h(t) = k_1e^{-.764t} + k_2e^{-5.24t} So now I must find the values for k_1 and...
  24. S

    Why is the characteristic of (d/dx) + (d/dt) = 0 not c = x + t?

    Why is the characteristic of (d/dx) + (d/dt) = 0 where d is small delta c = x - t and not c = x + t
  25. J

    Regular singular points of 2nd order ODE

    Homework Statement [PLAIN]http://img265.imageshack.us/img265/6778/complex.png I did the coefficient of the w' term. What about the w term? This seems like a fairly standard thing, but I can't seem to find it anywhere. What ansatz should I use for q, if the eqn is written w''+pw'+qw...
  26. D

    2nd order differential with limit - keep getting zero

    Homework Statement Find the solution of the following differential in the limit z -> i/2: Homework Equations Quotient rule, product rule. The Attempt at a Solution I found the first derivative using the quotient rule, with the answer as two fractions. I then took both fractions and...
  27. J

    Question about the def. of solving 2nd order ODEs through Var. of Parameters.

    Ok, so I've been studying the method of variation of parameters in order to solve 2nd order ODEs, and I have a question regarding a supposition that is made in the definition of the method. Say, y'' + p(t)y' + q(t)y = g(t) Then the general solution to the above equation is c_1y_1(t) +...
  28. M

    Splitting 2nd order DE into two 1st order DE's

    Homework Statement 1.a) Taking the Schrodinger second order differential equation given, split it into two first order differential equations for numerical solving. We're given the relevant constants in a table, such as \hbar, \alpha, etc. 'z' is taken to be an indepedent variable which we...
  29. A

    How Do You Solve a 2nd Order Inhomogeneous ODE with Given Initial Conditions?

    1. Using the complementary function and particular integral method find the solutio of the differential equation. d2y/dx^2 + 3 dy/dx +2y = 20cos2x Which satisfies y(0) = 1 y'(0) = 0 Homework Equations The Attempt at a Solution
  30. D

    Solving 2nd Order Nonlinear DE: Analyzing Effects of g Changes

    Hi everyone, I'm having a hard time analyzing the following problem: b v(x) = -exp(-x) - 1/2 ( g v'(x) )^2 - n x v'(x) + S(g) v''(x) where: v' = dv/dx, etc. 0 < b< 1 g > 0 n > 0 S(g) >0 and S'(g) >0 x \in (-inf, inf) The main goal is to figure out what happens as g...
  31. J

    2nd order non homogeneous diff. eqs. 2nd posting for clarification

    I recently attempted to solve the following: y” + (K/m)y = (Kl^{0}+mg)/m y(0) = l_{0} y(t_{e}) = (K l_{0}+mg)/K The Attempt at a Solution y(t) = -(mg/K)cos{\sqrt{K/m} t} + (mg/K){cos{\sqrt{K/m} t_{e}}/sin{\sqrt{K/m} t_{e}}}*sin{\sqrt{K/m} t} + (K l_{0}+mg)/K which...
  32. J

    2nd order diff eq, 3 dim. and wierd boundry condition

    I need help. For the following problem, can someone suggest how I should start on this question. I only have one quarter of diff eq classes plus a few classes in Fourier analysis. I'm out of my league. Consider a box with length, width and height given by L. The box encompasses the region...
  33. Y

    2nd order Boundary Value Problem.

    I want to solve: y(x)''-(\frac{m\pi}{a})^2y(x)=0 With boundary condition y(0)=y(a)=0. First part is very easy using constant coef. which give: y(x)=c_1 cosh(\frac{m\pi}{a}x) + c_2 sinh(\frac{m\pi}{a}x) y(0)=0 \;\Rightarrow\; c_1=0 \;\Rightarrow\; y(x) = c_2 \; sinh(\frac{m\pi}{a}...
  34. P

    2nd Order Runge-Kutta: 2nd Order Coupled Differential Equations

    Homework Statement Consider the system of coupled second-order differential equations u''-(t+1)(u')^2+2uv-u^3=\cos{t} 2v''+(\sin{t})u'v'-6u=2t+3 with initial conditions u(0)=1, u'(0)=2, v(0)=3, v'(0)=4. Use the second order Runge-Kutta method with h=0.2 and a=2/3, b=1/3, \alpha=\beta=3/2...
  35. A

    Solving a Coupled Trigonometric ODE System in MATLAB?

    So I have never solved a second order NON LINEAR ode before, but my current project has thrown one at me. I am going to use MATLAB to solve it when I have it in the correct state, but at the mo, I can't get it to that... Its three coulped equations, in terms of x,theta and x2. They are in a...
  36. S

    Solving 2nd Order ODEs w/ Variable Coefficients

    I want to know if there is a general solution to a second order homogeneous differential equation with variable coefficients?
  37. J

    2nd order differential equation

    I have the following equation d^2y/dx^2 +4dy/dx +8y = 4sin(2x) - 12cos(2x) y(0)=1 & y'(0)=-6 For the auxillary equation I have (m + 2)^2 = sqrt -4 which gives m=-2-2i & m=-2+2i Which gives y=e^-2x(Ccos(-2x) + Dsin(-2x)) Now to tackle particular integral. Try y=acos(2x) +...
  38. M

    How to Numerically Solve Coupled Second-Order ODEs Using RK4 in MATLAB?

    1. The problem statement. Two 2nd order ODEs are given as : x''-2xy''+x^2 y' = 0 x"+y"+xy'+yx'=0 write a code to solve them numerically using RK4 in matlab. The Attempt at a Solution I have some idea of starting this kind of problem. That is, I first need to convert them into Four 1st...
  39. B

    2nd order filter transfer function normalization

    I'm looking at a guide by Texas Instruments on active filter design. In it are the following equations for a second order lowpass filter, verbatim: The coefficient form of the denominator: s^2 + a_1s + a_0 Normalized: P(s) = (\frac{s}{\sqrt{a_0}*\omega_c})^2 + \frac{a_1s}{a_0*\omega_c} + 1...
  40. B

    Simplifying, two 2nd order linear differential equations

    I'm working on a computer programming project on matlab, and need an equation in the form of y'' + y' + y = 0 or k or inthe form of y' + y = 0 or k.. everything will be simple for me after that but I have these two equations N - ay'' - by' - cy - dx'' - ex' - fx = 0 M - gy'' - hy' - iy -...
  41. O

    A set of two 2nd order linear ode

    I'm doing an undergrad research job.. I have encountered the following coupled 2nd order linear ODE with constant coefficients a, b, c, d... \begin{align} \frac{d^2 y_1}{d x^2} + a^2y_1 & = -c y_2 \\ \frac{d^2 y_2}{d x^2} - b^2y_2 & = -d y_1 \end{align} In addition, I...
  42. K

    Can a Substitution Solve a Linear 2nd Order ODE with Non-Constant Coefficients?

    Homework Statement y''x^{2} + 4xy' -y = ln(x) The Attempt at a Solution -I considered the quadratic characteristic equation, but it won't work because of the x^2 -I also tried variation of parameters. so i have v = y' and v'=y'' but i have no idea what i would sub when I get to y. any...
  43. Q

    A simple 2nd order ODE using Green's functions

    Homework Statement Solve \frac{d^2 u(x)}{dx^2}=x(1-x) subject to the homogeneous boundary conditions u(0)=0=u(1), using Green's functions. Homework Equations Green's functions... The Attempt at a Solution There are three parts: i - homogeneous eqn: \frac{d^2...
  44. D

    Seeking example of 2nd order thermal system.

    I'm not very experienced on heat laws and devices, other than general thermodynamics. But I am studying 2nd order systems, and the curiosity came up if there is a 2nd order thermo system, since my book only described a 1st order one. I see the thermal capacity as an electronics capacitor...
  45. A

    2nd order non-homogeneneous ODE - how to find PI

    Homework Statement Find the general and, if possible, particular solutions of the following ordinary differential equations: y''+9y=36sin3x (hint: modification rule for PI) Homework Equations Knowledge of ODE's y = y_{aux}+y_{particular} The Attempt at a Solution I get the compementary...
  46. T

    System of ODE Boundary Value Problem with 2nd Order Backward Difference

    {\frac {{\it du}}{{\it dx}}}=998\,u+1998\,v {\frac {{\it dv}}{{\it dx}}}=-999\,u-1999\,v u \left( 0 \right) =1 v \left( 0 \right) =0 0<x<10 Second Order Backward Difference formula {\frac {f_{{k-2}}-4\,f_{{k-1}}+3f_{{k}}}{h}} I'm trying solve this numerically in matlab, but can't seem to...
  47. S

    Solving a 2nd Order Inhomogeneous ODE with Specific Boundary Conditions

    Homework Statement What's the general procedure for solving an i.v. b.v. ode of the form (d^2 T(x))/ (d x^2) = -c * e^T(x) T(+/-1) = 0 T'(0) = 0 where c is a constant Homework Equations i know from ode class that problems like this can usually be evaluated as T'' +...
  48. S

    Solving a 2nd Order Differential Equation with Exponential Terms

    Homework Statement \frac { d^2 \theta }{d x'^2 } = -y *exp(\theta) eq. 1 mayb be integrated to yield exp(\theta) = \frac {a}{cosh^2(b \frac{+}{-} \sqrt \frac{a*y}{2} * x')} \theta = f(y,x') Homework Equations The Attempt at a Solution the exponent is throwing me off, but i probably...
  49. J

    2nd Order Differential Equation Problem

    Homework Statement Let y be the solution of the initial value problem: y'' + 2y' + 2y = 0 , y(0) = 0 , y'(0) = 5 The maximum value of y over 0 less than or equal to x less than infinity is ??. The Attempt at a Solution r = -2 +/- i I solved it: y = c1 e^-2x cos(x) + c2 e^-2x sin(x) y' =...
  50. D

    Easy non-homogenous 2nd order diff.eq

    Homework Statement Find the general soulution to : y'' = sin(2x) and the particular solution that satisfies: y(0) = −1/4, y(pi/4) = pi/2. Homework Equations y = c1y1(x) + c2y2(x) + yp(x) The Attempt at a Solution y'' = sin2x ----> y = -sin(2x)/4 -0.25C1sin(2x) - 0.25C2sin(2x)...
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