What is Initial value problem: Definition and 178 Discussions

In multivariable calculus, an initial value problem[a] (ivp) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem.

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  1. F

    MHB Initial value problem with lots of constants

    L \frac{di}{dt}+Ri=E and we're given i(0)=i_o I,R,E,i_o are constants. So I rewrite equation as \frac{di}{dt}+\frac{R}{L}i=\frac{E}{L} therefore P(i)=\frac{R}{L} let \mu(x)=e^{\int \frac{R}{L}dt}=e^{\frac{tr}{L}+C} multiply equation by integrating factor to get e^{\frac{tR}{L}}...
  2. alane1994

    MHB Solving Initial Value Problem: \(\ln(t)+t^2y^2-\sin(y)=\pi^2\)

    Ok, I have a practice exam... My professor gave out a copy with worked out examples. There is one where I don't get his logic at all. I was wondering if you guys could explain it to me?\(\displaystyle (\frac{1}{t}+2y^2t)dt+(2yt^2-\cos(y))dy=0\) First, he put \(\text{Assume t>0}\) ...
  3. F

    MHB Solving the IVP for x=c_1\cos{t}+c_2\sin{t}: Solution to DE & Initial Conditions

    x=c_1\cos{t}+c_2\sin{t} is a two-parameter family of solutions of the DE x''+x=0 Find a solution of the IVP consisting of this differential equation and the following initial conditions: x(\frac{\pi}{6})=\frac{1}{2} and x'\frac{\pi}{6}=0 So x'=c_2\cos{t}-c_1\sin{t} x''=-c_2\sin{t}-c_1\cos{t}...
  4. A

    Inverse Laplace transform (Initial Value Problem)

    Homework Statement I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation): Y(s) = 1 / [ (s-1)^2 + 1 ]^2 The Attempt at a Solution I tried using a translation theorem and then apply the sine formula, but the denominator...
  5. C

    How can I solve a separable differential equation for an initial value problem?

    Homework Statement Solve the initial value problem: dx/dt = x(2-x) x(0) = 1 Homework Equations Problem statement. The Attempt at a Solution Based on the format, I attempted to solve the problem as a separable differential equation: ∫dx/(x[2-x]) = ∫dt Evaluating to...
  6. S

    Solve the initial value problem?

    Solve the initial value problem y'=2t(1+y), y(0)=0 by the method of successive approximations. I don't know how to do this problem but I think there's integral involved in it. Please help me. Thanks.
  7. djh101

    Finding the differential equation (initial value problem)

    Homework problem for nonlinear dynamics. Let us write xλ(t) for the solution of the initial value problem \dot{x} = f(x) & x(0) = λ where f is continuously differentiable on the whole line and f(0) = 0. a) Find the differential equation for \frac{∂x_{λ}}{∂λ}(t) I'm a little confused...
  8. MarkFL

    MHB Kavina's question at Yahoo Answers regarding an initial value problem

    Here is the question: Here is a link to the question: Calculus question on differential equations? - Yahoo! Answers I have posted a link there to this topic so the OP may find my response.
  9. E

    2nd order initial value problem in matlab

    Homework Statement 1.)I want to write a function in MATLAB that contains the 2nd order function: 20*d^{2}x;(dt^{2})+5*dx/dt + 20*x=0 (dampened spring) -The function should have 2 inputs (time,[initial values]) initial values should be a vector of 2 values -The function should...
  10. C

    Find a solution to the initial value problem

    Homework Statement Find a solution to the initial value problem that is continuous on the interval where Homework Equations I know the equations, but don't want to type them out.The Attempt at a Solution I got the first part of this question. The part where g(t) = sin(t) I can not figure...
  11. C

    Find the unique solution to the initial value problem

    Homework Statement The unique solution to the initial value problem is http://webwork.usi.edu/webwork2_files/tmp/equations/ed/12ad7dca5df62ed3b18f5fbf8c6e871.png Determine the constant and the function Homework Equations Not sure for the second part. The Attempt at a...
  12. D

    Second Order Differential Initial Value Problem

    Homework Statement y''+4y'+6y y(0) = 2; y'(0) = 4 Homework Equations \alpha ± β = e^{x\alpha}(cosβx + sinβx) The Attempt at a Solution Auxilary equation is r^2+4r+6, which solves for -2 ± i I get the general solution: e^{-2x}(c1cosx + c2sinx) y' = -2e^{-2x}(c1cosx +...
  13. C

    Can't finish a Laplace Initial Value Problem.

    I've had to take diff eqtns now and I'm trying to get my head around Laplace again.. it's been a while. I can't seem to transition to the simplest step of partial fractions, my denominators are tough to figure out. If someone could point me to the next step that'd be great! Thanks a lot guys...
  14. S

    Laplace Initial Value Problem with X and Y

    Hello. I have gotten as far as to use the Laplace equation with these formulas, but I am having difficulty getting y and x to relate to each other. If requested, I can post my work, but I am sure it is fraught with mistakes. Help is very much appreciated! x' + 2y' - x - 2y = e^t x' - y' + x...
  15. C

    Initial value problem with y(0)=C

    Hey everyone, I'm a long-time visitor, it's my first time posting though. I have a homework problem that is causing me considerable consternation: (y^3)*(dy/dx)=(8y^4+14)*cos(x); y(0)=C Oh, and we're supposed to solve the initial-value problem, and then solve for the particular...
  16. E

    Unique Solution for IVP of Continuous Functions with Global Lipschitz Condition

    Homework Statement For the space of continuous functions C[0,T] suppose we have the metric ρ(x,y) =sup _{t\in [0,T]}e^{-Lt}\left|x(t)-y(t)\right| for T>0, L≥0. Consider the IVP problem given by x'(t) = f(t,x(t)) for t >0, x(0) = x_{0} Where f: ℝ×ℝ→ℝ is continuous and globally Lipschitz...
  17. I

    Solve the following initial value problem (DiracDelta function)

    Homework Statement y'' +4y = 2 delta(t - pi/4) where y(0)=0 and y'(0)=0 Homework Equations Laplace transform Inverse Laplace transform The Attempt at a Solution after applying laplace tranform Y(s)=2e^((-pi/4)*s) / s^(2)+4 as the final answer i have y(t) =...
  18. J

    Initial value problem, first order linear ode

    Hey, We haven't properly covered this in class yet, but I am trying to study ahead using online course notes, I manage to finish a few questions but I have gotten stuck here, The question starts by asking for the solution to the ODE: y' = 1 - 2xy, When I solve this using the...
  19. M

    Help solving the Initial Value Problem

    Homework Statement y''-4y'+4y=0 , y(1)=1 and y'(1)=1 The Attempt at a Solution Auxiliary equation: r2-4r+4=0 I tried factoring 2 different ways: (r-2)2=0 r=2,r=2 y1=e2t y2=y1 y(t)=c1e2t+c2e2t y(1)=c1e2+c2e2=1 ---eq(1)y'(t)=2c1e2t+2c2e2t ...c2=1/(2e2)-c1 ---eq(2) sub eq(2) into eq(1)...
  20. M

    Help solving the initial value problem

    Given: Solve the initial value problem 2(√x)y'+y+4(√x) ; y(1)=2 I am having trouble separating the x's and y's in order to integrate. I keep coming up with: dy/dx +y/(2(√x))=2... What do I keep missing here? I am pretty sure you leave the y(1)=2 alone until you are finished with...
  21. M

    Initial Value Problem. I'm really confused just need some help

    Homework Statement Solve the following Initial Value problem for x(t) and give the value of x(1) Homework Equations (dx/dt)-xt=-t , x(0)=2 The Attempt at a Solution (dx/dt)-xt = -t (dx/dt) = xt-t (dx/dt) = t(x-1) (1/(x-1)) (dx/dt) = t (1/(x-1)) dx = t dt Then I integrate...
  22. F

    How to Solve the Initial Value Problem with Laplace Transforms?

    Homework Statement Solve IVP y'' + 2y' + 2y = u_pi(t) + u_2pi(t) with IC y(0) = 0 and y'(0) = 1. Homework Equations L{f''(t)} = s^2Y(s) - sy(0) - y'(0) u_c(t) = u(t-c) -->Laplace--> e^-cs/s The Attempt at a Solution y'' + 2y' + 2y = u_pi(t) + u_2pi(t) y'' + 2y' + 2y...
  23. O

    MHB What Methods Solve This Initial Value ODE?

    how can we solve this ODE? http://img818.imageshack.us/img818/3966/59962234.png
  24. E

    Initial Value Problem / Differential Equation

    Thanks for clicking! So, I've got a problem here that I'm stuck on. I need to find the general solution to y' = (y3 + 6y2 + 9y)/9 I found this to be ln|y| + (3/(y+3)) - ln|y+3| = x + c but I would appreciate it if you would check my work. Anywho, once I have the general solution I...
  25. R

    Solving the Initial Value Problem for 'y' = x, x' = -5y-4x

    y' = x x' = -5y-4x y(0) = 1 x(0) = 0 after finding the general solution as shown here http://www.wolframalpha.com/input/?i=y%27+%3D+x%2C+x%27+%3D+-5y-4x how do you go about applying the initial values and finding the complete solution?
  26. MathWarrior

    Initial Value Problem Differential Equation

    L\frac{dI}{dt}+RI=E I(0) = I_{0} Where E is a constant. I know I need to separate the equation and integrate but I am not quite sure how given all the variables running around... I don't see how the condition of I(0) = I_{0} helps in any way.
  27. M

    Need help finding general solution of an initial value problem

    The equation is y'' + 4y' + 4y = (3 + x)e-x and initial conditions y(0) = 2, y'(0)=5so from the associated homogenous equation I think the fundamental set of solutions is {e^-2x, xe^-2x} and so yc would be Yc = c1e-2x + c2xe-2x but now I don't know how to get Yp, particular solution or what...
  28. D

    Solution of Initial Value Problem

    Homework Statement Determine the solution of the IVP y' + 4ty = 4t, y(0) = 6 Homework Equations The Attempt at a Solution p(t) = 4t g(t) = 4t μ(t) = e^{\int4tdt} = e^{\int p(t)} = e^{\int4tdt} = e^{2t^{2}} is this all I need? because i did \frac{d}{dt}(y * μ(t)) = p(t)...
  29. L

    Solve the Initial Value problem

    dy/dt=t^(2)y^(3) , y(0)=-1 I need help solving this I put the integral (dy/y^3)= integral (t^2)dt but idk what to do after that or if that's even right
  30. F

    Initial Value Problem (done but i think its wrong please check workthanks)

    Homework Statement R(dQ/dt) + (1/C)Q = E_0 e^-t ...Q(0) = 0 and E_0 = a constant Homework Equations The Attempt at a Solution first i rearranged to give: Q' + (1/CR)Q = (E_0e^-t)/R next i multiplied all by integrating factor of: u(t) = e^integ:(1/CR) = e^(t/CR)...
  31. C

    Initial value problem, end of problem Q

    Homework Statement Solve the I.V.P. x2(dy/dx) = (4x2-x-2)/((x+1)(y+1)) , y(1)=1 Homework Equations The Attempt at a Solution So far, I got to this: y2/2 + y = log(x) + 2/x + 3log(x+1) + C I used the initial conditions to solve for C and got: C = -1/2 - 3log(2) Substituting C...
  32. C

    Help with initial value problem

    OK, so clearly I am missing something, because I know this is supposed to be a simple problem. It reads: solve the following initial value problem: dy/dt=-y+5 y(0)=y_naught my process is as follows: dy/(5-y)=dt integrate ln(5-y)=t+C exponential both sides 5-y=(e^t)(e^c)...
  33. A

    Initial value problem and laplace transform

    Homework Statement I understand how to do initial value problems but I'm slightly stuck when the initial values are y(0) = y'(0)=0 The question is Solve: y''+3y''+2y=f(t), y(0)=y'(0)=0 where f(t) is a square wave. Homework Equations \Im{y'} =s\Im{y}-y(0)...
  34. C

    Hard power series and initial value problem question

    Homework Statement We know that y = Aex is the solution to the initial value problem dy/dx = y; y(0) = A. This can be shown by solving the equation directly. The goal of this problem is to reach the same conclusion using power series. Method: Let y be a solution to the initial value...
  35. G

    Initial value problem of ord diff eq

    Homework Statement 3y'' -y' + (x+1)y = 1 y(0) = y'(0) = 0 Homework Equations Not sure, that's the issue The Attempt at a Solution I can't quite get this one using the methods I'm familiar with, and I can't guess a particular solution to neither the equation nor the...
  36. L

    Initial value problem Euler equation

    Question: Find y as a function of x: x^2 y'' + 8 x y' - 18 y = x^8 y(1)=3, y'(1)=2 Attempted solution: I found the general equation to be Ax^(-9)+Bx^2+Cx^8. However when I try to solve the initial value problem for this equation I have 3 unknowns.
  37. L

    Strategies for Solving Differential Equations Missing the Dependent Variable

    I'm given the following DE and initial conditions: y''=2yy' y(0)=0, y'(0)=1 I started by doing a reduction of order like so: w=y', w'=y'', \int w=y=\frac{w^{2}}{2}+c which then gave me this: w'=2w(\frac{w^{2}}{2}+c) w'=w^{3}+2wc Now I'm stuck on where to go from here. I can't use any of the...
  38. M

    Find the solution to the initial value problem

    Homework Statement Find the solution to the initial value problem dy/dx - y = e^3x y(0) = 3 Homework Equations e^∫p(x) The Attempt at a Solution Do I treat p(x) = -1? I(x) = e^∫-1 = e^-x e^-x(dy/dx) - ye^-x = e^3x . e^-x e^-x(dy/dx) - e^-x . y = e^2x e^-x . y = ∫e^2x y = (2e^2x...
  39. R

    Solving an Initial value problem using Laplace transform

    Homework Statement Solve the Initial value problem using Laplace transform \ddot{y} +2y = 0, y0 = C1, \dot{y} = C2 Homework Equations [s2 - sy(0) - y'(0)] + a[sY - y(0)] + bY The Attempt at a Solution s2Y - sy(0) - y'(0) + 2y = 0 s2Y + 2Y = sy(0) + y'(0) (s2 + 2)Y = s(C1) + (C2)...
  40. 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 =...
  41. N

    Is My Approach to Solving the 3rd Order IVP Correct?

    a 3rd order IVP I am havin trouble with: y''' -3y'' +2y' = t + e^t y(0)=1, y'(0)= -.25 y''(0)= -1.5 I am using At^2 and B*e^t *t as my Y1 and Y2. Is this correct?
  42. M

    Find solution of initial value problem - 1st order non-linear ODE

    Hey, we have to solve the following problem for our ODE class. Homework Statement Find the solution of the initial value problem dx/dt = (x^2 + t*x - t^2)/t^2 with t≠0 , x(t_0) = x_0 Describe the (maximal) domain of definition of the solution. The Attempt at a Solution Well...
  43. S

    Laplace Transform Initial Value Problem

    1. y'' -5y' + 6y = 0, y(0) = 1, y'(0) =2 3. [s^2 F(s) - s f(0) - f'(0)] -5 [F(s) - f(0)] + 6[F(s)] = 0 (s^2 +1)F(s) - (s -5)f(0) - f'(0) = 0 (s^2 + 1)F(s) - (s-5)(1) - 2 = 0 (s^2 + 1)F(s) = s -3 F(s) = (s-3)/(s^2 + 1) Here's where I'm stuck. I can't factor the denominator to...
  44. N

    Resolve Initial Value Problem | Find y0 for Diverging Solutions

    I am having trouble with the below problem: y'-(3/2)y= 3t+ 2e^t, y(0)= y0. fine value of y0 that separate solutions that grow positively and negatively as t=> infinity. I found p(t) to be -3/2, u(t) to be e^-3t/2 => e^-3t/2*y' - 3y/2( e^-3t/2)= e^-3t/2(3t+ 2e^t) => -2 -4e^t + ce^...
  45. N

    How Do You Solve an Initial Value Problem with Integration Steps?

    I need help with an initial value problem, ty' + (t+1)y= t; y (LN 2)= 1 I divided t and have u(t) as exp Integral of t+1/1 => e^t +t Multiplied this to the original equation to get (e^t +t)y' + ((t+ 1)/t) *y *(e^t +t) = (e^t +t) How can I integrate the above? Are my steps so far...
  46. B

    Initial value Problem, ODE

    Homework Statement Proof that there exist more than one solution to following equation \frac{dx}{dt} = \sqrt[3]{x^{2}} , x(0) = 0Homework Equations The Attempt at a Solution Well, I need a confirmation to my attempt of solution. The one is quite forward: \Rightarrow x=(1/3(t+c))^{3} Pluging...
  47. P

    General solution to system of equations and initial value problem

    Homework Statement Find the general solution for the following systems of equations, a solution to the initial value problem and plot the phase portrait. --> this is in matrix formx' = 1 2 0 3 all multiplied by x. also, x(0) = 2 -1 Homework Equations Determinant, etc.The Attempt at...
  48. K

    Initial Value Problem using Laplace

    Is this problem possible? Solve the initial value problem x''(t) + 6x'(t) + 9x(t) = f(t); x(0) = N, x'(0) = M I get to X(s)=(F(t)+Ns+6N+M)/(s^2+15) and don't know where to go from here. Any help would be appreciated.
  49. C

    Solving IVP without Initial Equation: Step-by-Step Guide

    Hey, I need some guidance on an IVP. In general, how do you proceed on these types of problems when you have only the initial values but no initial equation? For example, I have x1(0)=1 and x2(0)=0 but that is it. I understand, for example, how to do IVP's in the context of separating...
  50. E

    Solve the initial value problem explicitly for u

    Homework Statement solve the initial value problem for u du/dt= (2t + sec^2(t))/2u also, u(0)=4 Homework Equations antiderivative of sec^2(t) is tan(t) + C The Attempt at a Solution So, the first thing i did was move the "u" with the "u" and "t" with the "t". so the equation looks like...
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