2nd order Definition and 494 Threads

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  1. J

    Proof of the linearity principle for a 2nd order PDE?

    Homework Statement My textbook (Advanced Engineering Mathematics, seventh edition, Kreyszig) indicates that if u1 and u2 are solutions to a second-order homogeneous partial differential equation, and c1 and c2 are constants, then u where u = c1u1 + c2u2 is also a solution, this is the...
  2. T

    How Does Unity Position Feedback Affect System Dynamics in Control Theory?

    Homework Statement With unity position feedbck, i.e. make K2=0, plot root locus as a function of pitch gain (K1). By imposing 2nd order system approximation, estimate settling time, rise time, peak time of the closed-loop system with 20% overshoot. Pic of system...
  3. K

    Why does the 2nd order homogeneous linear ODE have 2 general solutions?

    why not the 2nd order linear homogeneous ODEs have three Linearly independent solutions or more? I know for the characteristic equation, we can only find 2 answers but.. just wondering if that is the only case to solve the question and if it is, then why it has to be. so my question is,1. 2nd...
  4. D

    2nd order differential equation: undetermined coefficients

    Homework Statement y''-y=t-4e^(-t)Homework Equations method of undetermined coefficients The Attempt at a Solution solving for characteristic equation first y''-y=0 r^2-1=0 c_1e^(-t)+c_2e^(t) RHS particular solution t-4e^(-t) y_p(t)= At+B+Ce^(-t) y_pt'(t)=A-Ce^(-t) y_p''(t)=Ce^(-t)...
  5. genxium

    Function whose 2nd order divergence is the Dirac Delta

    Homework Statement This problem came when I was learning the Poisson's equation (refer to http://farside.ph.utexas.edu/teaching/em/lectures/node31.html). when it came to the step to find the Green's function G which satisfies \nabla^2 \cdot G(\textbf{r}, \textbf{r}') =...
  6. B

    What form for the particular integral of this 2nd order Diff Eqn

    Hi Folks, I have the following forced torsional vibration problem of the form ##\displaystyle J_0 \ddot{\theta}+k_t\theta=\frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos n w t+b_n \sin n w t)## I assume the solution of the CF is in the form ##\theta=A \cos nwt+B\sin nwt## but I am not sure what...
  7. M

    Solving for nonlinear 2nd order DE

    Given a DE in the general form of either y'' = y^2 or y'' = (y-1)^2, is there a general method to solve these? I separated the equations to get y''(y^-2) = 1 and then integrated, which left me with (-y^-1)dy = (t + c)dt, and then integrated once more. Is this correct so far? I have essentially...
  8. M

    2nd order, linear, homogeneous proof

    Hi, I was wondering if someone could provide either a bit of intuition or a mathematical proof (or both) as to why if the Wronskian (W(f,g)) does not equal 0 for all t in an interval, then the linear combinations of the two functions f and g encompass ALL solutions. Is there any particular...
  9. DavitosanX

    Homogeneous 2nd Order DE from spring pendulum

    I'm currently taking a Classical Mechanics course, and we're studying the lagrangian equation. After a few exercises, I thought I'd try to come up with the motion equations for a pendulum where the mass hangs from a spring. The resulting differential equations take a form that I don't really...
  10. M

    2nd order DE w/ various solutions

    Hi, I'm currently finding a solution (i.e. y) for y′′ +t(y′)2 = 0. The answer is: y=(1/k)ln|(k−t)/(k+t)|+c2 if c1 =k^2 >0; y=(2/k)arctan(t/k)+c2 if c1 =−k^2 <0; y=−2t−1 +c2 if c1 =0; also y=c (NOTE: c1 or c2 is just c_1 or c_2) I've gotten the solutions as y = 0 or y=(2/k)arctan(t/k)+c2...
  11. F

    MHB Help on a 2nd Order Linear Differential Equation

    Hi, I need help solving this ODE. I know the answer is a Bessel function but I need help on the process of getting there. Initial conditions y(0)=1 and y'(0)=0 xy''+y'+xy=0 I have made it this far... ${x}^{2}*\sum_{n=0}^{inf} [n(n+1)*{C}_{n+2}+(n+1)*{C}_{n+1}+{C}_{n}]*{x}^{n}=0$
  12. F

    MHB Need Help Solving a 2nd Order Nonlinear Differential Equation

    Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] I am pretty much lost on how to go about solving this. It would be greatly appreciated if someone could point me in the right direction on this. Thanks!
  13. F

    2nd Order Nonlinear ODE Question

    Homework Statement Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] It would be greatly appreciated if someone could point me in the right direction on this. Thanks! Homework Equations The Attempt at a...
  14. J

    2nd Order Nonlinear Differential Equation

    Hi everyone, I need some help to solve this differential equation. The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system: diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 " Currently, I am still...
  15. T

    Change of variables for 2nd order differential

    Hi there, I'm having some difficulty in understanding how the change of variables by considering a retarded time frame can be obtained for this particular eqn I have. Say I have this original equation, \frac{\partial A}{\partial z} + \beta_1 \frac{\partial A}{\partial t}+ \frac{i...
  16. T

    Is My Interpretation of the 2nd Order Derivative Correct?

    Hi there, I'm kind of rusty on some stuff, so hope someone can help enlighten me. I have an expression E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z] I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to...
  17. C

    Significance of standard form of 1st and 2nd order transfer functions?

    I foolishly skipped most of my analogue electronics classes, and inevitably failed the exam. I'm now trying to revise for the resit but I'm so far behind that I just cannot understand any of the lecture slides, and I'm getting very stressed. The part of the module I am revising at the moment...
  18. Greg Bernhardt

    What are Nonlinear Optics - 2nd Order Processes

    Definition/Summary Nonlinear optical processes that occur due to the presence of a second-order nonlinear susceptibility are termed 2nd order processes, or three-wave mixing processes. There are four second order processes, second harmonic generation, sum and difference frequency generation...
  19. C

    Non linear 2nd order differential equation

    please provide step by step method to solve this 2nd order non linear differential equation: attached with this thread. take FUCOS(Ѡt) and FUsin(Ѡt) as zero.
  20. P

    Solving Second Order Nonhomogeneous DEs with Initial Conditions

    Homework Statement Homework Equations y=yPI+yCF The Attempt at a Solution First issue is I was under the impression that a particular solution is the final solution to a DE; a solved DE with initial conditions applied, but it would be weird for that to be the first part of the...
  21. D

    Nonhomogeneous 2nd order dif equation

    Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am...
  22. N

    Solving 2nd order PDE of single variable

    I've been getting pretty rusty in terms of derivation in recent years. Encountered this problem which I can't derive the steps despite knowing the solution. \frac{\partial^2 u}{\partial r^2} + \frac{\partial u}{\partial r}\left(\beta + \frac{1}{r}\right)+\frac{\beta}{r}u=0 Known...
  23. K

    Laplace Transform solution for 2nd order differential equation

    Homework Statement d^2x/dt^2 - 3 dx/dt + 2x= 2e^3t give that at t=0, x=5, and dx/dt=7 Homework Equations i can't figure out how to derive the values of A, B, and C from the attempted equation solution. please help me out here. thanks The Attempt at a Solution
  24. J

    Is PID Control Possible for a 2nd Order System?

    PID of "2nd order"? Exist PID control of "2nd order"? Ie., a command system of correction to error that includes a factor of correction proportional to 2nd derivative, another proportional to 1nd derivative, another proportional directly to error, another proportional to 1nd integral and...
  25. F

    2nd Order Differential Equation with Improved Euler Method (Heun's)

    Homework Statement I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct...
  26. gfd43tg

    Damping Coefficient and Resonant Frequency of a 2nd order circuit

    Hello, I am having major confusion for how to find the damping coefficient, α, and the resonance frequency, ωo, for a second order circuit, in general. I know that there are tables for standard circuits like a series RLC and parallel RLC giving those values, but I am certain that I need to know...
  27. C

    2nd order ODE - undetermined coefficients?

    Homework Statement 1) Find the general solution of y''+ω02=Ccos3(ωx) 2) Show there exists two frequencies at which resonance occurs and determine them The Attempt at a Solution I've tried the method of undetermined coefficients, assuming a solution of the form y=(Acos(ωx)+Bsin(ωx))3...
  28. C

    Variable coefficient 2nd order DE

    Homework Statement This an example from Boyce+DiPrima's text on ODEs. Original problem is to solve 2t2y''+3ty'-y=0, given that y1=1/t is a solution. But I'm stuck at the part where I'm to solve 2tv''-v'=0, v being the second unknown solution. The Attempt at a Solution In the text...
  29. quasar987

    2nd order homogenous constant coefficient ODE question

    If we have a constant coefficient second order homogeneous ODE, the way to solve this is to suppose a solution of the exponential type. This yields a second order polynomial equation (the "characteristic equation") that the exponent must satisfy. In case the solutions of the characteristic...
  30. MarkFL

    MHB Solve Linear Inhomogeneous 2nd Order IVP: Joe Smalls' Question

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  31. MarkFL

    MHB Solving 2nd Order Homogeneous ODE - Joe's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  32. J

    Coupled 2nd order diff eq's (Bessel functions?)

    I have derived these pair of coupled diff equations for U_1 (r) and U_2 (r): r^2 \dfrac{d^2 U_1 (r)}{dr^2} + r \dfrac{d U_1 (r)}{dr} + r^2 U_2(r) = 0 and r^2 \dfrac{d^2 U_2 (r)}{dr^2} + r \dfrac{d U_2 (r)}{dr} - r^2 U_1(r) = 0 Or written in matrix form (r^2 \dfrac{d^2}{dr^2} + r...
  33. jimmy neutron

    Poincare's theorem and 2nd order ode

    This is a request about the second order differential equation y'' + (k^2 + f(r))y(r) = 0 (1) where k is a (real) constant and f(r) is a real valued function of r that has some constraints regarding integratability. According to...
  34. M

    How Do You Solve a 2nd Order Linear Non-Homogeneous DE with Imaginary Roots?

    when you solve a 2nd order linear non-homogeneous DE, where it is equal to a constant as in Kirchoff's 2nd Law and the roots of the auxiliary equation are imaginary then you have superposition of 2 solutions. so the particular solution is equal to a constant k and you can solve for this by...
  35. C

    MHB How to Solve This 2nd Order ODE in Control Systems?

    I'm trying to solve this equation analytically, but I can't even find the auxiliary equation or general solution! Km = 0.5 C*e = 0 K2 = 0.03 K1 = 0.05 x* = 49
  36. MarkFL

    MHB Solve 2nd Order Linear Inhomogeneous ODE: Muhammad Fasih

    Here is the question: I have posted a link there to this thread so the OP may see my work.
  37. M

    Calculate 2nd order Taylor polynomial of a given function

    Homework Statement . Let ##f:\mathbb R^2 \to \mathbb R## / ##f \in C^2##, ##f(0,0)=0## and ##\nabla f(0,0)=(0,1)## ##Df(0,0)=\begin{pmatrix} 1 & 1\\ 1 & 2\\ \end{pmatrix}## Let ##g:\mathbb...
  38. F

    "2nd Order to Matrix: Find Fundamental Matrix

    Homework Statement x'' + 3x' + 2x = 0 Find fundamental matrix Homework Equations x = x1 x' = x2 = x1' x'' = x3 = x2' = x1'' The Attempt at a Solution Not sure how to convert this to a matrix... The eiganvalues should be 1 and 2
  39. H

    Variation of parameters- 2nd order linear equation

    Homework Statement solve 4y''-4y'+y=16et/2 Homework Equations v1= -∫ y2g/w v2= ∫ y1g/w The Attempt at a Solution http://imgur.com/gxXlfdH the correct answer is 2t^2 e^(t/2) instead of what i have though, i am not sure what i am doing wrong?
  40. I

    Solving 2nd Order Differential Equation with Initial Conditions

    Mod note: Reinstated problem after poster deleted it. Homework Statement Just wondering if I did this correctly: ##y''+4y'+4y=e^{x}## and initial conditions ##y(0)=0; y'(0)=1## Homework Equations The Attempt at a Solution So I found the characteristic equation to be...
  41. M

    2nd Order Perturbation Coefficients.

    I have found an expression for the estimated energy contribution a term |I> will bring to a wavefunction |K> \Delta E = \frac{|\langle I|\hat{H}| K\rangle|^2}{(E_K - \langle I |\hat{H}| I\rangle)} Is there a simple way to extract the coefficient that will be associated with |I>? Even a link...
  42. P

    2nd order PDE using integration by parts

    Homework Statement Find the general solution of the equation (\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0, where ##\zeta## and ##\eta## are independent variables. Homework Equations The Attempt at a Solution I set ##X = \partial u/\partial\eta## so that...
  43. MarkFL

    MHB Solve 2nd Order Inhomogeneous ODE: Kartika's Q on Yahoo Answers

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  44. G

    Series soln. with singular point to 2nd order linear differential eqn

    So, I'm trying to solve 2nd order linear differential equations (series solutions near a singular point). (lnx)y" + 0.5y' + y = 0 around the regular singular point x = 1 I got the indicial equation, r(r-0.5) = 0, which leads to the roots... r1 = 0.5, r2 = 0 The problem...
  45. G

    Series solutions near a singular point, 2nd order linear

    So, I'm trying to solve 2nd order linear differential equations (series solutions near a singular point). (lnx)y" + 0.5y' + y = 0 around the regular singular point x = 1 I got the indicial equation, r(r-0.5) = 0, which leads to the roots... r1 = 0.5, r2 = 0 The problem only asks us...
  46. V

    2nd Order Differential Equation via Power Series

    Homework Statement Problem 5 on the attached Sheet here Homework Equations We studied the Power Series Method and how to calculate a linearly independent solution if one solution is already known. So we need to find one solution (probably) using the power series method and then using...
  47. T

    Need help with solving an 2nd Order Linear ODE

    Homework Statement $$\frac { { d }^{ 2 }x }{ d{ t }^{ 2 } } +{ w }^{ 2 }x={ F }_{ 0 }sinwt\quad \quad \quad \quad x(0)=0\quad \quad x'(0)=0$$ Homework Equations The Attempt at a Solution $$\frac { { d }^{ 2 }x }{ d{ t }^{ 2 } } +{ w }^{ 2 }x={ F }_{ 0 }sinwt\quad \quad \quad \quad...
  48. P

    2nd order ODE's odd and even functions

    Hi I have been looking at some lecture notes inc the following example. Solve : y'' + ω^2y = (some even function) The particular integral is then found using Fourier series. As the function on the RHS is even this only includes cosine terms. The complementary function is found from...
  49. T

    How Can One 2nd Order ODE Have Different Solutions?

    2nd Order ODE "Contradiction"? To solve a 2nd order ODE, we can follow the steps as shown below. (Image 2 is a continuation from Image 1, apologies for the size difference.) :rolleyes: The method to obtain the solution is straightforward. Let's say \frac{d^2y}{dx^2}=ky If k = -1, a...
  50. E

    Optimal Controller Gain for Desired Response in 2nd Order Laplace Transforms

    Homework Statement Solve the DE for y(t) with the IC's y(0)=20.8m/s and y'(0)=0 if the input is a step function scaled by the desired velocity Vo. vd(t)=Vou(t). Assume the desired velocity Vo=27.8m/s Homework Equations y''(t) + (D/M)y'(t) + (K/M)y(t) = (K/M)vd(t) M = 1,000kg D = 100kg/s K...
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