What is Ivp: Definition and 144 Discussions

Homework Statement Find the solution of the given initial value problem. \[y''-2y'-3y=3te^{2t}\]Homework Equations The Attempt at a Solution(1) the homogenous solution is given by \[y_{h}=c_{1}e^{3t}+c_{2}e^{-t}\] (2) the particular solution is in the form \[y_{p}=(At+B)e^{2t}\] (3) The...

Homework Statement This is more theoretical than anything, but let's say that a problem is stated like: Consider the IVP: 'given DE'; y(0) = y_0 find the value of y_0 that separates the solutions that grow positively from those that grow negatively as t -> infinity. The Attempt at a...

1. Find the first four nonzero terms of the power series approximation of the solution. y"-4y = 4t-8e-2t y(0)=1, y'(0)=-1 2. y=\suma_n*t^n where the summation goes from 0 to infinity 3. I have done a homogeneous problem similar to this and had no problems finding the first four...

Thanks for the help. Homework Statement Solve the IVP: x'=-3x+4y-2, y'=-2x+3y, x(o)=-1, y(o)=3 The Attempt at a Solution x''=-3x'+4y' x''=-3x+4(-2x+3y)=-3x-8x+12y y=(x'+3x+2)/4 x''=-3x-8x+12((x'+3x+2)/4) x''-x=6 xgeneral=xh+xp Xg=Ce^t+Ce^-t-6 Yg=1/4(Ce^t-Ce^-t+3Ce^t+3Ce^t)...

Homework Statement Solve the IVP for the wave equation: Utt-Uxx=0 for t>0 U=0 for t=0 Ut=[dirac(x+1)-dirac(x-1)] for t=0 2. The attempt at a solution By D' Almbert's solution: 1/2 integral [dirac(x+1)-dirac(x-1)] dx from (x-t) to (x+t) I apologize for not using Latex- my...

Homework Statement "Use Laplace transforms to solve the initial value problem" from section 4.2 in Elementary Differential Equations (6th ed.) Edwards & Penny x'' + x = cos3t; x(0) = 1 & x'(0) = 0 Homework Equations L{coskt} = s/(s²+k²) Apparently the answer is...

dy/dx = 3[(y-1)^(1/3)] with initial values: y(0)=1 ultimately I end up with y=sqrt((2x)^3) + 1 or ((2x)^3/2) + 1 book answer: y= 1+((3x)^3/2) steps: separation of variables... ((y-1)^-1/3)dy = 3dx after u-subs, where u = y-1... 3/2((y-1)^2/3) = 3x + C...

Homework Statement Show that the DE y'=\sqrt{y} has more than one solution when y(0) = 0 by finding two of them. Why does Picard's uniqueness theorem not apply? The Attempt at a Solution By standard ODE techniques, y=\frac{1}{4}[2c+c^2+t^2] but if y(0)=0 then y=\frac{t^2}{4} only...

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...

Homework Statement suppose that h \inC(R) and f\inC(R^2) satisfies |f(t,x)|\leqh(t) |x| for all (t,x) \inR^2. Show that for any point , the IVP : x' = f( t, x) x( \tau)=\varsigma has a solution which exists for all t in R Homework Equations The Attempt at a Solution

Homework Statement y'' + ty' - y = 0 plugging in the known Laplace formuals i get this... [s^2Y(s) - sy(0) - y'(0)] + [-sY'(s) - Y(s)] - Y(s) = 0 Homework Equations y(0) = 0 y'(0) = 3 The Attempt at a Solution simplying to a linear first order DE -sY'(s) + (s^2-2)Y(s) = 3...

Homework Statement Consider the initial value problem \begin{align*} \left\{ \begin{array}{l} \displaystyle \frac{dy}{dx} = \exp(xy) \\ y(0) = 1 \end{array} \right. \end{align*} 1. Verify that this IVP has a unique solution in a neighborhood of x = 0. 2. Following the...

Homework Statement Find all solutions of the IVP y'' + a(t)y' + b(t)y = 0, y(t0) = 0, y'(t0) = 0 where t0 is any fixed point on the t-axis and the coefficients are continuous. The Attempt at a Solution I know this has to do with the Existence and Uniqueness theorem. How would I apply...

Homework Statement I'm trying to use the Laplace transform to find the charging equation of a capacitor with an initial voltage already on the capacitor.Homework Equations V_i = RC\frac{dVc}{dt} + V_c The initial condition of Vc will be \lambda. The Attempt at a Solution V_i(s) = sRC*V_c(s)...

Tricky problem. Any tips? Thanks SOO much! :biggrin: Homework Statement Let f be continuous on [a,b] and let c be a real number. If for every x in [a,b] f(x) is NOT c, then either f(x) > c for all x in [a,b] OR f(x) < c for all x in [a,b]. Prove this using a) Bolzano-Wierstrass and b)...

Homework Statement mass1 = 10kg connected to ceiling by a spring mass2 = 2kg connected to mass1 by a spring 2m/s is applied to mass 1 in the downward direction (positive direction) to get the system in motion At equilibrium Mass 1 is 2m from the ceiling while mass 2 is 4m from the...

x' = Ax, x(0) = v, where A is the matrix in problem 6 and v = [1,2]^T. Do not use the eigenvalues and the eigenvectors of A. (Hint A^2v = 0).A = [ (1, -1)^T (1, -1)^T ] All I did was calculate e^(tA)v = v[I + tA]. In this case, an IVP, x(t) is the solution. x(t) = e^(tA)v. Since I wasn't...

Homework Statement The solution to the Initial value problem, x''+2x'+5x=0, is the sum of the steady periodic solution x_sp and x_tr. Find both. Homework Equations The Attempt at a Solution I already found x_sp ( the particular solution). It is (-44/533)cos(7t)+(14/533)sin(7t). r^2+2r+5=0...

Homework Statement Find the series solution to the initial value problem. xy\acute{}\acute{} + y\acute{} + 2y = 0 y(1) = 2 y\acute{}(1) = 4 Homework Equations y=\sum^{\infty}_{n=0}c_{n}(x-1)^{n} t = (x-1), x = (t+1) y = \sum^{\infty}_{n=0}c_{n}t^{n} y\acute{}=...

Homework Statement I am given the task for finding the laplace transformation given y'' + 4y = 8t^2 if 0<t<5 and 0 if t>5 y(1) = 1+cos(2) y'(1)= 4-2sin(2) Homework Equations The Attempt at a Solution I know that the above problem can be written y''+4y=8 \cdot u(t) where u(t) is the step...

I had to solve an IVP: \partial/u\partialt + 4\partialu/\partialx = e2x, u(x,0) = f(x). I got an answer of u = 1/2 +f(x-4t). Is this correct and if not, where did I go wrong.

Homework Statement Find y as a function of t if 36y''-132y'+121y=0, y(0)=5, y'(0)=4 The Attempt at a Solution 36y''-132y'+121y=0 36r^2-132r+121=0 (6r-11)^2 So, general solution y(x) = C1*e^(11x/6)+C2*x*e^(11x/6) y'(x)=(11/6)*C1*e^(11x/6)+C2*e^(11x/6)*((6x-11)-(36/121)) y(0)= C1=5 y'(0)=...

Homework Statement Solve the IVP and determine if the solution is unique and explain why. Homework Equations t*y'+(t-2)y=t^4*e^t, y(0)=0 The Attempt at a Solution t*y'+(t-2)y=t^4*e^t y'+ ((t-2)/t)y=t^3*e^t integration factor=e^(integral((t-2)/t dt)=e^t/(t^2)...

Homework Statement Use the Laplace Transform to solve: y"+2y'+2y=t y(0)=y'(0)=1Homework Equations L{y(t)} = Y(s) L{y'(t)} = sY(s)-y(s) L{y"(t)} = s2Y(s)-sy(0)-y'(0) using the laplace transform table: tn = n!/(sn+1) where n=1The Attempt at a Solution Take laplace on both sides: L{y"(t)} +...

Okay so suppose I have the Initial Value Problem: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{IVP} NB. I am considering only real functions of real variables. If f(x,y) is...

I just took an exam and the final problem was the following: dy/dx = (y-3) e^(cos(xy)) y(1)=1 prove that y(x) < 3 for all x for all y that are defined. To be honest, I tried a few things but realized I didn't know where to start. I can show that the limit of the slope from neg...

Hi, I am a little bit confused about singular solutions and their relationship with IVP's and decided to ask you. As far as I understood, the IVP's could be in a form: y' = f(x,y) y(x0) = y0 To obtain a general solution, we could use separable eq. method. I have learned that...

Homework Statement y" +4y = g(x) ; y (0) = 1 & y(0) = 2 g(x) = sinx , 0 \leqx \leq pi/2 0 , x \succ pi/2 Homework Equations The Attempt at a Solution 1> g(x) = sin x solving i get : yc = ec1sin2x + ec2cos2x US set of sin x = {sinx...

Hi all. Could someone give me a hand with this: Using Euler's method find an approximate solution to the IVP using step size h = 0.1 \frac{dy}{dx} = 4xy + 3 , y(0)=0 I know how to use Euler's method for something simple like \frac{dy}{dx} = x + y But I'm just not to sure what...

Homework Statement Solve the IVP for t>0 y'(t)+\int^{t}_{0}y(\tau)d\tau = =sin(t) for 0<t<=pi =0 for t>pi y(0)=-1 The Attempt at a Solution The solution depends on how large t is and therefore the solution consist of two parts depending on the size of t.. Let's call them y1(t)...

Use the variable change u = ln(y) to solve the IVP dy/dt = -y ln(y), y(1) = 2? We haven't covered this in class yet so I do not know where to even start.

Homework Statement Solve the following IVP for 1st order quasilinear PDE s using the method of characteristics. u*u_x + y*u_y = x u = 2s, y = s, x = s Homework Equations a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z) z = u(x_o,y_o) = 2s The Attempt at a Solution The...

[SOLVED] IVP and DE problem Homework Statement 1) Find a solution to the IVP y" = -9y, y'(0) = 3, and y(0) = 6 2) Is y(x) = (21+6x)^1/3 a solution to the DE y' = 2/y^2 The Attempt at a Solution For the first one I think it has something to do with sin/cos but I don't know what to do...

pde : du/dx*du/dy = xy IC: u(x,y) = x for y = 0

Homework Statement Please take a look at my work and help me figure out where I went wrong. Thanks! Use separation of variables to solve the IVP \[ty'=\sqrt{1-y^2},\quad y(1)=1,\quad t>0 Homework EquationsThe Attempt at a Solution Use separation of variables to solve the IVP...

Homework Statement y''+4y=t^2+3e^t y(0)=0 y'(0)=2 Homework Equations CE: r^2+4 r=+/-2i gs: y=c1 cos(2t) + c2 sin(2t) The Attempt at a Solution guess: yp=(At^2+Bt+C)e^t yp'=At^2e^t+2Ate^t+Bte^t+Be^t+Ce^t yp''=At^2e^t+4Ate^t+Bte^t+2Ae^t+2Be^t+Ce^t back into problem...

Okay, I know this is alot... but I am stuck, so here goes... Use the method of Laplace transform to solve the initial value problem y''+3ty'-6y=0, y(0) = 1, y'(0) = 0 L\{y'' + 3ty' - 6y\} = L\{0\} s^{2}Y(s) - sy(0) - y'(0) + 3L\{ty'\} - 6Y(s) = 0 s^{2}Y(s) - s(1) - 0 -...

Higher Order Homogeneous ODE (IVP) [Solved] I am having problems with this IVP: y'''' + y' = 0 y(0) = 5 y'(0) = 2 y''(0) = 4 What I have done so far is: \lambda^3 + \lambda = 0 \lambda(\lambda^2 + 1) = 0 So one roots is \lambda = 0 (though.. can there be a root that...

I have a system of spatial ODEs to solve... Actually a DAE system, but here's the issue: The equations are vaild over a specific domain, x = 0..L The equations are only bound at one point (thier "initial point") but at either 0 or L f1(0)=0 f2(0)=100 f3(L)=0 f4(L)=100 (also an...

Q. Determine whether the initial value problem y' = x^5 - y^5 + 2xe^y ,y\left( 3 \right) = \pi has a unique solution. I've seen things like Lipschitz bound and conditions on the partial derivative of f(x,y) with respect to y where y' = f(x,y) but such things mean very little to me. The...

Ok I'm stuck on this problem - and it didnt particularly look hard when I started doing it, oh say about a good 2 hours ago.. hmm Solve the IVP 2y'' + 3y' + y = t^2 + 85 sin(2t), y(0) = -2, y'(0) = 0. My homogeneous solution is c1y1 + c2y2 y1 = e^\frac{-t}{2}, y2 = e^-^t, Wronskian =...

there's an example in the text that we're supposed to use to solve this problem. the example solves the ODE u" = f, & to find the fundamental solution F(x) we want to solve F"(x) = \delta (x) where \delta (x) is the Dirac delta function. the Heaviside function satisfies (H(x)+c)' = delta(x) for...

I'm trying to solve an IVP problem but have not had a lot of Diff Eq to really understand this. The IVP is dx/dt = -(1-(1/x))*(1/(x^.5)) , x>1 x(0) = 2 I guess some of the things I'm looking for with this is: Any unique solutions equilibrium points Initial velocity distance...

Hi! I have :eek: hand in an assigment in like four hours and part of it involves solving a IVP. We have to use the built in function ODE45. the system of equations are as follows. x'(t)=-axy y'(t)=axy-by z'(t)=by Use step length 1 and solve the ivp numerically for the following values...