2nd order Definition and 464 Threads

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...

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)...

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}') =...

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...

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...

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...

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...

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...

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... ${x}^{2}*\sum_{n=0}^{inf} [n(n+1)*{C}_{n+2}+(n+1)*{C}_{n+1}+{C}_{n}]*{x}^{n}=0$

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!

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...

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...

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...

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...

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...

[SIZE="4"]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...

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.

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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

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

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...

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...

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

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

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...

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

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?

Mod note: Reinstated problem after poster deleted it. [/color] 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...

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...

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...

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

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...

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...

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...

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...

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...

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...

[b]1. Check that y(t)=1/λ ∫_0-t_〖f(s) *sin(λ(t-s) )ds〗 is the solution of the following initial value problem y''(t)+λ^2y(t)=f(t), λ>0, y(0)=0,y'(0)=0 Homework Equations [b]3. I tried to do integration by parts on y(t), but...

While trying to solve a problem in physics I got this system of 3, second order differential equations. Solution should be equation for linear harmonic oscillator. m\ddot{η}1+kη1-kη2=0 mn\ddot{η}2-kη1+2kη2-kη3=0 m\ddot{η}3-kη2+kη3=0 My attempts at the solution produced 6th order...

can someone help me solve the differential equation that takes the following form? y''+Ay'+By+Cy^2=f(x), y is function of x Thanks a lot!

Hi, everyone! This is my first post here, I need an hand with this equation! Homework Statement Solve the initial value problem: \begin{equation} \begin{cases} u''(x)+4u(x)=\cos(2x) \\u(0)=u'(0)=1 \end{cases} \end{equation} The Attempt at a Solution I started by solving the...