General solution Definition and 303 Threads

Here's the situation: A small block of wood of inertia mb is released from rest a distance h above the ground, directly above your head. You decide to shoot it with your pellet gun, which fires a pellet of inertia mp. After the block has fallen a distance d, the pellet hits it and becomes...

Homework Statement Find the general solution for the DE: t2y''-2y=0 Homework Equations These were given for other parts of the problem so I'm not sure if they're relevant. y1(t)=t2, y2(t)=t-1, y(1)=-2, y'(1)=-7 The Attempt at a Solution The t2 at the front was really stumping me and I'm not...

Homework Statement [/B] (dy/dx)^2 = (1-y^2) / (1-x^2)Homework Equations Separating the variables I arrive with: dy/sqrt(1-y^2) = dx/sqrt(1-x^2) By integration on both sides by trigonometric substitution and putting it in a general solution: Arcsin y - Arcsin x = C The Attempt at a Solution If...

Homework Statement Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0. Homework Equations None. The Attempt at a Solution I'll post my whole work.

Homework Statement Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0. Homework Equations None. The Attempt at a Solution I got up to an+2=-an(n+3)/(n+1) for n=1, 2, 3, 4, 5, 6... a3=-2a1 a4=0 a5=3a1 a6=0 a7=-4a1 a8=0 The answer in the book says y=a0sigma from m=0 to...

So, I have been trying to come up with a general solution for dI/dt in an RLC circuit. I have attached the work I have done so far. I don't know where but I am making a mistake and the waveform is not coming out right. Would really appreciate a look over my work to see if I made any obvious errors.

Question : General solution of dy/dt = -ay + b My solution : dy/dt = -a(y-b/a) (dy/dt)/(y-(b/a)) = -a Integrating both sides : ln | y-(b/a) | = -at + C e(-at+C) = y-(b/a) Ce(-at) = y-(b/a)

Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions. How does one prove that...

I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...

Hey! :o I want to solve the differential equation $y'''(x)-2y''(x)+y'(x)=e^x$. I have done the following: Consider the homogeneous equation $y'''(x)-2y''(x)+y'(x)$. $k^3-2k^2+k=0 \Rightarrow k=0 \text{ single root } , k=1 \text{ double root } $ So, the solution of the homogeneous...

any particular solution plus the general solution to the homogeneous equation. I'm having difficuilty understanding this proof from my lecture notes Theorem : Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w T(v) = 0. where v ∈ V (the kernel ) to prove: T(u) = w...

Hello There, I hope I posted this in the right thread. I've been struggling with solving this particular Differential Equation and just couldn't find any way to solve it completely.. 1. Homework Statement I am only required to find the general solution of the differential equation y'' + y' =...

We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states: $$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$ and $$ \psi_k(x)= \begin{cases} A(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx})...

Homework Statement Find the general solution to differential equation y''-6y'+9y=e^x((2x+1)\cos x+(x+3)\sin x) Homework Equations -Non homogeneous differential equation -Homogeneous differential equation with constant coefficients -Method of undetermined coefficients The Attempt at a Solution...

Hi, in the book 'Introduction to Quantum Mechanics' by Griffiths, on page 71 in the section 'The Delta-Function Potential' he states that the general solution to time independent Schrodinger Equation is $$\psi(x) = Ae^{-\kappa x} + B e^{\kappa x}$$ he then notes that the first term blows up as...

I am having a little trouble with a problem I am trying to solve. Given three particular solutions Y1(x)= 1, Y2(x)= x and Y3(x)= x^2 Write down a general solution to the second order non homogeneous differential equation. What I have done so far is to realize if Y1,2 and 3 are solutions...

Just a question about the theory of solutions to differential equations? Given a second order differential equation and two particular solutions y1 and y2, what is the best way to find the general solution? i.e variation of parameters or something else

Homework Statement By making the transformation u= x^αy where α is a constant to be found, find the general solution of[/B] y'' + (2/x)y' + 9y=0The Attempt at a Solution I've worked out y,y',y'' and subbed them into get x^-au'' + x^a-1(2-2a)u' + x^-a-2(x^2-a(a-1))u =0...

Hi everyone I'm taking a linear algebra class at university right now and this is one of my homework questions . I am unsure how to even approach these questions. Any pointers in the right direction would be greatly appreciated. I apologize in advance for not showing any attempt at this...

1. The problem statement Given a stochastic matrix P with states s_1...s_5: P = \begin{pmatrix} 1 & p_2 & 0 & 0 & 0\\ 0 & 0 & p_3 & 0 & 0\\ 0 & q_2 & 0 & p_4 & 0\\ 0 & 0 & q_3 & 0 & 0 \\ 0 & 0 & 0 & q_4 & 1 \end{pmatrix} and the matrix A (which is obviously related to P, but I can't see...

I'm supposed to use undetermined coefficients to find a general solution to: y" + 4y' +3y =4^(-t) I can't find an example online where f(t) is equal to an exponential function that does not have e as the base, so I have no idea how to solve it. So far, I found the general solution to the...

I herewith submit for reference, review and comment the general solution to 3 simultaneous equations as follows:

Hi all, My name is Arijit Biswas. I have resumed learning maths after a long time and I am stuck with a simple problem in trigonometry. I need to find a general solution to the equation: sec2 2x = 1– tan 2x. I have worked out something i.e. 1) Multiply by cos2 2x and that makes the equation...

Homework Statement Find general solution of the given problem in the picHomework Equations All I need to know is where they found the fundamental set of solutions. The Attempt at a Solution All I need to know is where they found the fundamental set of solutions. Please see pic. All I want...

Hello! (Wave)I found the following in my lecture notes:$$u_t=u_{xx}, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x)$$ $$u(x,t)=X(x)T(t)$$ $$\Rightarrow \frac{T'(t)}{T(t)}=\frac{X''(x)}{X(x)}=-\lambda \in \mathbb{R}$$ $$X''(x)+\lambda X(x)=0, x \in \mathbb{R}$$ $$X \text{ bounded }$$ The characteristic...

Hello! I have two uncertainties (hehe) about two concepts from a QM time-dependent Schrödinger equation video. The video is I cannot move on further if I don't fully grasp everything he explains in the video. My two issues are: 1) The general solution for the time-dependent Schrödinger...

Hi everyone, I am trying to find the general solution for the following ode: y'' +gy' + 10y = e2xcos(x) The solution states that the answer is y = 1/145 (5cos(x) + 2sin(x))e2x + (Acos(x) + Bsin(x))e-3x I was able to correctly find the homogeneous part of the equation as e-3x (Acos(x) +...

2nd order ODE has a form y''+p(x)y'+q(x)y=f(x)and if we assume f(x)=/=0 for every x, then y''+p(x)y'+q(x)y=/=0 so in this case we can't specify general solution of 2nd order ode?

Homework Statement If ##u(x,t)\in \mathbb R, x\in[-\pi,\pi] ## represents a standing wave with ##u(\pm\pi,t)=0## Then what is the most general solution u(x,t)? Homework EquationsThe Attempt at a Solution [/B] Using the separation of variable technique: u(x,t)=P(x)Q(t) I get ##P(x)...

Getting stuck on something I think that could be trivial. Maybe someone can see my mistake. consider the system: $x' = -2x + y$ and $y' = 2x - 3y$ a) Write the system in matrix form my solution $\overrightarrow{X} = (^x_y)$ so: $X' = (^{x'}_{y'})$ so $A = $ \begin{bmatrix} -2 & 1 \\ 2 & -3...

Homework Statement (1+x2) y'' + 2xy' = 0 in powers of x Homework Equations y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} y' = \sum_{n=1}^{\infty} na_nx^{n-1} The Attempt at a Solution (1+x2) y'' + 2xy' = (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1}...

Hello, I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation. It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by: Ygeneral = Yhomogeneous + Yparticular My question: Is it...

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of **A**. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f)...

Hey! :o Find the solution of the problem $$u_{tt}(x, t)-u_{xx}(x, t)=0, 0<x<\pi, t>0 \tag {*} \\ u(0, t)=0, t>0 \\ u_x(\pi ,t)=-u_{tt}(\pi ,t), t>0$$ I have done the following: We are looking for solutions of the form $$u(x, t)=X(x) \cdot T(t)$$ $$u(0, t)=X(0) \cdot T(t)=0 \Rightarrow...

Hi, This is overwhelmingly more of a maths problem than a physics problem, because it's all theoretical. I'll give some background to modle it incase the math's isn't enough. Say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0...

Find the general solution of the given differential equation. $y'' + 2y' + y = 2e^{-t}$ I understand how to do everything in the problem- but I do not understand how we are obtaining $y = At^2e^{-t}$? Is this just some formula I should memorize or how exactly am I supposed to guess or figure...

\frac {dy}{dt} = y^3 + t^2, y(0) = 0 My teacher said this IVP couldn't be expressed in terms of functions we commonly know. I was wondering what the general solution is? Thank-you

Find the general solution of the given differential equation.. $$(1+t^2)y' + 4ty = (1+t^2)^{-2}$$ I'm kind of confused here on what to do... Do I want to do something like $$e ^{\int4t} dt$$ and then multiply that through on both sides or do I need to do something different here..I'm not...

Homework Statement Find the general solution. Homework Equations y"+y=x2sin2x The Attempt at a Solution Characteristic equation would be: m2 + 1 = 0 So,m2 = -1 Therefore, m = i or m = -i. Complementary function would be : Asinx+Bcosx where,A and B are constants respectively. If I write...

While fiddling around with some very simple linear ODEs, I "discovered" a formula that gives a solution to ODEs of the form: ##y'+y=ax^n ##. here it is: i'm sure that this was discovered before, but i was just wondering if it had any official name or something.

Suppose that we have the four-vector potential of the electromagnetic field, [texA^i[/tex] The wave equation is given by $$(\frac {1}{c^2} \frac {\partial^2}{\partial t^2}-\nabla^2) A^i=0$$ Now the solution, for a purely spatial potential vector, is given by $$\mathbf{A}(t...

Hi, Problem: Given the DE: y^(6) + y = 0, find the general solution. Solution. I found the roots to be: (sqrt3 - i)/2, (sqrt3 + i)/2, i, -i, (-sqrt3 + i)/2, (-sqrt3 - i)/2 Thus, my general solution led to: y = c1cost + c2sint + c3exp(t/2*(sqrt3))*cos(t/2) + c4exp(t/2*(sqrt3))*sin(t/2) +...

Tell me if the following is correct. For a simple infinite square well potential, the solutions to the Schrodinger equation are \Psi_n(x)=\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a}), then you plug in the appropriate value for n and operate on the function accordingly to get your observables. Then...

Homework Statement I'm trying to figure out how the general solution of the Schrodinger equation for a free particle when v=0 relates to anything I have learned in class...Homework Equations For Eψ=(hbar2/2m)d2ψ/dx2The Attempt at a Solution I really have no idea- what is confusing me is that ψ...

Homework Statement Find the general solution of u_{xx} + u = 6y, in terms of arbitrary functions.Homework Equations The PDE has the homogeneous solution, u(x,y)=Acos(x)+Bsin(x) . u_{xx} + u = 6y has the particular solution, u(x,y)=6y The Attempt at a Solution Taking a superposition...

Homework Statement From Mary Boas' "Mathematical Methods in the Physical Science" 3rd Edition Chapter 3 Sec 11 Problem 33 ( 3.11.33 ). Find the eigenvalues and the eigenvectors of the real symmetric matrix. $$M=\begin{pmatrix} A & H \\ H & B \end{pmatrix}$$ Show the eigenvalues are real and...

Homework Statement for this question , i dun know which method should i use... can someone enlighten me on this? i sub y=vx then differentiate with respect to x but can't get the ans Homework Equations The Attempt at a Solution

Homework Statement i got stucked here. below is the answer given. can anybody help please? Homework Equations The Attempt at a Solution

Homework Statement i can't reduce the eq to y= (X+1)/(x+c ) , which part is wrong? Homework Equations The Attempt at a Solution

Homework Statement Find the general solution of y'=Ay. Your answer must be a real-valued function. A= \begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix} Homework Equations The Attempt at a Solution The first step would be to find the eigenvalues. I forgot the name of the term but if...