What is Second order ode: Definition and 91 Discussions

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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  1. M

    Solving 2nd Order ODEs: y^4 -3y'' -4y = 0

    Hi. I am new to differential equations. This is probably pretty easy but I don't quite understand how to do it yet. The equation is y^4 -3y'' -4y = 0. I can figure out what class of equation it is. I can write it in the form y'' = F(y), but I am not really sure how to solve it.
  2. A

    How to solve this non-homogenous second order ODE?

    Homework Statement y''-2y+y=xe^xlnx The Attempt at a Solution I don't know what I should do here because lnx. Is it possible to solve this ODE with undetermined coefficients method? how can I solve it?
  3. L

    Solving 2nd Order ODE: du/ds & k^2

    d^{2}u/ds^{2}= cosu[(du/ds)^{2} - k^{2}]
  4. T

    Break Second order ODE into a system of first order ODE's

    Homework Statement I haven't done this for several years and have forgotten. Kicking myself now over it since it looks like something so simple but i cannot figure it out... I need to break this second order ODE into a system of first order ODE's in matrix form to use within a crank...
  5. M

    Second order ODE initial value problem

    So the question is y" - y' - 6y = e^-x + 12x, y(0)=1,y'(0)=-2 First I found the general solution which came out to be, Ae^3x + Be^-2x I then Substituted y=ae^-x + bx + c y'=-ae^-x + b y"=ae^-x Then I just compared the coefficients to get a=-1/4, B=-2 and C=-1/6 So I am getting y =...
  6. U

    Euler Bernoulli to second order ode

    hello, I have read in a number of papers that if we have a cantilever beam and are only interested in the movement of the tip when the base is being excited at the frequency of the first eigen mode , then the whole beam can be replaced by a spring mass system. Can anyone tell me of a...
  7. C

    Trying to solve a second order ODE

    Homework Statement I'm trying to solve a second order ODE for y(x) to show that the solution is y(x)=sin(x)/x. We've been told to use the substitution y(x)=h(x)/x. I've got to the stage of solving for h(x), arriving at h''(x)=-x. Using the general solution, h(x)=A sin(x) + B cos(x) and...
  8. C

    Second order ODE application question

    We are doing mass spring problems that stem from second order ODE's. I think my lack of linear algebra is hurting me once again in this section so any help would be greatly appreciated. We are using a stiffness matrix of K = [ -(k1+k2), k2 (row 2) k2, -(k2+k3)] Our first problem has the...
  9. S

    Ode45 to solve nonlinear second order ode

    Homework Statement I'm given two equations first (d^2)*r/dt^2 - r((d*theta/dt)^2)= (-A)/r^2 --- this is a non linear second order differential equation second (r^2)*((d*theta)/dt)=B B and A are...
  10. M

    Reducing Second Order ODE system to First Order

    Homework Statement A 3-storey building can be modeled as a system of coupled masses and springs as showen in attached document. Where mi is the mass of each floor, ki is the spring constant, xi is the displacement of each floor, and ci is the damping coeffcient.Homework Equations I understand...
  11. 0

    Particular Solution of an Inhomogeneous Second Order ODE

    Homework Statement A particular solution of y'' + 4y = tanx Answer choices are: (a) 1/2*cos(2x)ln|sec(2x)+tan(2x)| (b) -1/2*cos(2x)ln|sec(2x)+tan(2x)| (c) 1/2*sin(2x)(ln*cos(x)+x*sec(2x)) (d) 1/2*sin(2x)(ln*cos(x)-x*sec(2x)) (e) none of the above Homework Equations The...
  12. 0

    Inhomogeneous Second Order ODE

    Homework Statement What is the value of a such that the solution of the initial-value problem satisfies limx->infinity y(x) = 0? y''+y'=e^(-x), y(0)=1, y'(0)=a Homework Equations The Attempt at a Solution Not sure what to do with the missing y term... yp=Ae^(-x), y'p=-A^(-x), y''p=A^(-x)...
  13. J

    Inhomogeneous second order ODE with non-constant coefficient

    Homework Statement Solve ODE of form y''+(2/x)y'=C*(e^y) where C is a constant Homework Equations The Attempt at a Solution I don't really see how to approach this one, so a point in the right direction would be great. Thanks,
  14. J

    Error function as a solution to a second order ode

    Hi I need to find the solution of d^2y/dx^2 + 2x(dy/dx) = 0 I've solved it in Maple and get that y=a*erf(x)+b but I have no idea how to arrive at this answer! Any help would be great, thanks.
  15. P

    Bessel function Solution to Second order ODE with exponential coefficient

    Homework Statement Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2 Homework Equations - The Attempt at a Solution First I did a change of variables: Let u = e^(-t) Then du/dt = -e^(-t) dx/dt = dx/du*du/dt = -e^(-t)*dx/du d2x/dt2 = d/du(dx/dt)du/dt =...
  16. M

    How Does the Wronskian Affect Linear Independence in Second Order ODEs?

    y1 and y2 are solutions to the ODE L[y]=0=y''+p(x)y'+q(x)yWhat can you conclude about p(x), q(x) and the solutions on the interval I if i) W(x) = 0 for all X on I ii) W(x) = c for all X on I, c =/= 0 --- W(x) = y_1'y_2-y_1y_2' = C*e^{\int{p(x)}} i) W=0 so y1'y2=y1y2' And y1 and y2 are...
  17. D

    Second Order ODE with Weird Coefficients: Solving the Equation x2-2x+1

    Homework Statement y''(x-1)-xy'+y=x2-2x+1 The Attempt at a Solution I don't even know how to start this, Don't know what substitution to do.
  18. O

    Solving 2nd Order ODE: Even Function Solution

    Could you please help me or give me any hint to solve this ODE.. \frac{d^2y}{d x^2} + ( 2\rm{sech}^2 x - a^2 ) y = 0 where a is a constant. I want only even function solution. (y(x) = y(-x)) BTW, this is a homework problem. I encountered this equation while considering surface waves...
  19. C

    Nonhomogeneous Second Order ODE containing log

    Hi guy, I have this ODE that I'm having problems with y"+4y'+4y= e^(-2x)logx Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before
  20. S

    Reducing Order of ODE Ly with Ansatz y2: Find u to Solve

    Ly ≡ (x +1)^2y′′− 4(x +1)y′+6y =0 given y[1]=(x+1)^2 is a solution, use the ansatz y2(x)= u(x)(x+1)2 to reduce the order of the differential equation and find a second independent solution y2 how to reduce !? and i can't find u ...can't solve (x+1)^2u''+6u=0 please help! thx!
  21. B

    Solving non-linear second order ODE

    Homework Statement It will be great if someone could show me some options: I need to prove the following: A particle of mass m is attracted toward a fixed point O (the origin) with a force proportional to its instantaneous distance from O raised to a positive integer power, i.e...
  22. N

    Solving 2nd Order ODE for k: 20dx²/dt² + dx/dt + kx = 0

    Find k: 20\frac{d^2x}{dt^2} + \frac{dx}{dt} + kx = 0 when t = 0 - x = 1 - dx/dt = 1 My attempt at this in maxima: (%i1) 20*'diff( 'diff(x, t), t) + 'diff(x, t) + k*x = 0; 2 d x dx (%o1) 20 --- + -- +...
  23. M

    How Do You Solve Autonomous Second Order ODEs Like y'' = f(y)?

    Hi All, I was looking for the general solution of an equation as y(x)'' = f (y), and found the attached document on the web. It presents the solution in a way which I am not sure I understand. I tried to look at the trivila example y'' = - y, solution y = sin (x), but I am struggling in...
  24. B

    Solving a Second Order ODE with V(t) = -Q/C

    Homework Statement Question and part 1 as above. The second part involves solving this equation where L = 8R^2 C. The system is kept in steady state by maintaining V(t) = -Q/C (constant). V(t) is then set to 0 at t=0. It also says "Note that V(t)=0 for t>0 and that appropriate initial...
  25. S

    Solving 2nd Order ODE: y''+(1/x)y'=0

    Homework Statement find the general solution to the ODE: y''+(1/x)y'=0 Homework Equations The Attempt at a Solution I put this in the following form: y''=-(1/x)y' integrated both sides: y'=-ln(x)y +C I think i made a stupid mistake but i can't figure out what it is. Any help...
  26. T

    Second Order ODE - Variation of Parameters

    Homework Statement Find the general solution of the following diff. eqn. y''(t) + 4y'(t) + 4y(t) = t^(-2)*e^(-2t) where t>0 Homework Equations General soln - Φgeneral(t) + Φparticular(t) Wronskian - Φ1(t)Φ22'(t) - Φ2(t)Φ1'(t) The Attempt at a Solution I'm solving by...
  27. K

    Second order ODE solution for this system?

    second order ODE solution for this system?? hello guys, I am wondering if what is the analytical solution for this system? can we solve it as a mass-spring-damper system? thanks for your helps. the rectangular part is removed from the disk. http://img3.imageshack.us/img3/3610/odev.th.jpg
  28. J

    Second Order ODE Using Laplace Transforms

    Little homework problem that I'm beating myself up over... Solve: xy'' - 2y' + xy = -2\cos x Using the method of Laplace transforms... So I do some jiggling and get to: (1+s^{2})\frac{dF}{ds}+4sF(s) = \frac{2s}{s^{2} + 1} To which I find the following solution: F(s) =...
  29. B

    Tips for Solving a 1D Second Order ODE in an Open Channel Experiment

    Hi all, My maths are very rusty and I would need some advice. I have some experimental results obtained in an open channel and got depth-averaged velocity u(y) at different cross-sectional locations y. I tried different models but there is one I don't know how to tackle. The following one...
  30. J

    Help in second order ode numeric solution

    Dear all, Im trying to solve the following ode: y'' = -0.12*y + 0.4/sqrt(y^2 + 5.76) , y=y(t) , t: [-50,50] y(-50)=2.3 , y'(-50)=0 i changed it to a set of two first order ode using z=y' and solve it with finite differences. note that the right side...
  31. K

    Why is the solution for a second order ODE -m_{l}^{2} e^{im_{l}\phi}?

    Can someone explain to me why the solution of \frac{d^{2}\Phi (\phi)}{d\phi^{2}} = -m_{l}^{2} is \Phi = e^{im_{l}\phi}?
  32. R

    How to Solve a First and Second Order ODE for y(x) and u=dy/dx

    The Problem You are given: http://img530.imageshack.us/img530/4468/88346476ca9.jpg Where http://img408.imageshack.us/img408/209/53113174nt5.jpg is constant (taken as B). a) Differentiate both sides to produce a second order ODE for y(x) b) Show that it can be written as a first order ODE...
  33. T

    Is y(t) = c1t^2 + c2 t^−1 the general solution of a second order ODE?

    Hi, I am trying to decide whether y(t) = c1t^2 + c2 t^−1, where c1 and c2 are arbitrary constants, is the general solution of the differential equation (t^2)y'' − 2y = 0 for t > 0 and justify the answer, but I don't really know how to approach it from this "side" of the problem. Any suggestions...
  34. I

    Second order ode with non constant coeffcients

    Homework Statement y''(x)-k y^2 y'(x)=0 The Attempt at a Solution mathematica gives me this...
  35. J

    Second Order ODE - Initial Value Problem

    Solve the initial value problem y''+3y'+2y = 3e^{2t}+1 with initial values y(0) = 1, y'(0) = 1. I am unsure if I am going about the solution correctly. 1.) Find the characteristic equation. r^{2}+3r+2=0 \Rightarrow (r + 1)(r + 2) = 0 Therefore, y = c1•e^{-t}+c2•e^{-2t} 2.) Use method of...
  36. K

    MATLAB Having a problem with solving a second order ODE equation using Matlab

    I am having a problem coding a Matlab code that solves a second ODE equation which I give below: x^3*(1-2*x*M)d^2J(x)/dx^2+2*(2*x^2+i*nu*x-7*x^3*M)*dJ(x)/dx -2*(2*x+8*M*x^2+i*nu)*J(x)=0. where M = 1 (Mass of a black hole), nu = 0.74734+0.17792*i, J is a function of x, i represents a...
  37. K

    MATLAB Having a problem with soling a second order ODE equation using Matlab

    I am having a problem coding a Matlab code that solves a second ODE equation which I give below: x^3*(1-2*x*M)d^2J(x)/dx^2+2*(2*x^2+i*nu*x-7*x^3*M)*dJ(x)/dx -2*(2*x+8*M*x^2+i*nu)*J(x)=0. where M = 1 (Mass of a black hole), nu = 0.74734+0.17792*i, J is a function of x, i represents a complex...
  38. S

    Characterizing second order ODE

    Regarding: (a+bx+cx^2)y^{''}+(f+gx+hx^2)y^{'}+(j+kx+mx^2)y=0 Does anyone here know if it's been "completely" characterized in terms of the geometry of the three parabolas which make up it's coefficients? For example, if I'm given plots of the parabolas, can any information at all be...
  39. Clausius2

    Linear Independence of Solutions in Second Order ODEs

    Assume the next differential LINEAR second order equation: w''+\frac{4}{x}w'+\frac{4}{x^4}w=0 So I thought: OK, I need two independent solutions w_1 and w_2, because the space of solutions is of dimension two. Then the professor gave us a solution: w=sen(2/x)-(2/x)cos(2/x) and I...
  40. E

    How can I combine the solutions for u(t) and y(t) to find the solution for y(t)?

    Suppose we have y'' = f(t, y); y(a) = y0; y'(a) = y0' Note all derivatives are with respect to t. Let u = y', then u' = y'' 1. u' = f(t, y), u(a) = y'(a) 2. y' = u, y(a) = y0 Question 1: For y' = u, should I think of this as dy/du = u? Otherwise, I don't see how to solve 2 because...
  41. N

    Solving 2nd Order ODE: r\ddot\theta-g\sin\theta=0

    How would I go about finding a solution to this differential equation? r\ddot\theta-g\sin\theta=0 Where r and g are constants.
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