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. MathMan2022

    Finding the maximum size of an Initial Value Problem coefficient

    The following IVP diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39 Describes the tempetatur T in celcius at the time x of a tub filled with water. A tub which is filled with hot water at rate of v l/min. Lets say I am told that a guy takes a 40 min bath, and during those 40...
  2. A

    I Non-linear ODE: initial conditions

    Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...
  3. karush

    MHB -2.4.2 interval of initial value problem

    Determine an interval in which the solution of the given initial value problem is certain to exist $t(t-4)y'+y=0 \quad y(2)=2\quad 0<t<4$ ok my first step was isolate y' s $y'=-\dfrac{y}{t(t-4)}$ not sure what direction to go since we are concerned about an interval
  4. T

    ADM formulation Initial Value Problem data per spacepoint

    I'm having a bit of trouble getting a clear picture of what is going on here, so if anyone can shed any light, it will be greatly appreciated. 1. I can see how the metric coefficients provide the six numbers per spacepoint, but it can't always be possible to transform the metric into a diagonal...
  5. karush

    MHB Solving an Initial Value Problem Analytically

    Solve the initial value problem $y'=\dfrac{1+3x^2}{3y^2-6y}, \quad y(0)=1$ Solving analytically $3y^2-6y\ dy = 1+3x^2 \ dx$ so far hopefully...
  6. karush

    MHB DE 2.1.1.16 Find the solution of the give initial value problem

    Find the solution of the give initial value problem $\displaystyle y^\prime - \frac{2}{t}y =\frac{\cos{t}}{t^2}; \quad y{(\pi)}=0, \quad t>0$$u(t)=e^{2 \ln{t}}$then $\displaystyle e^{2\ln{t}}\, y^\prime - \frac{2e^{e^{2\ln{t}}}}{t}y = \frac{e^{2\ln{t}}\cos{t}}{t^2}$not sure actually!
  7. karush

    MHB 31.6 Solve the initial value problem

    $\tiny{31.6}$ Solve the initial value problem $Y'=\left|\begin{array}{rr}2 & 1 \\-1 & 2 \end{array}\right|Y +\left|\begin{array}{rr}e^x \\0 \end{array}\right|, \quad Y(0)=\left|\begin{array}{rr} 1 \\1 \end{array}\right| $ ok so we have the form $y'=AY+G$ rewrite as $$\displaystyle...
  8. karush

    MHB 3.1.11 find the solution of the given initial value problem:

    find the solution of the given initial value problem: $6y''-5y'+y=0\quad y(0)=4 \quad y'(0)=0$ if $r=e^{5t}$ then $\displaystyle 6y''-5y'+y=(r-3)(r-2)=0$ then $y=c_1e^{3t}+c_1e^{2t}=0$ for $y(0)=4$ $y(0)=c_1e^{3(0)}+c_1e^{2(0)}=4$ ok I don't see how the last few steps lead to the...
  9. V

    Solve the initial value problem

    Homework Statement Solve the initial value problem y1'=-13y1+4y2 y2'=-24y1+7y2 y1(0)=5, y2(0)=2 Homework EquationsThe Attempt at a Solution Here is what I have: y'=[-13 4, -24 7]y I change it to A=[-13 4, -24 7] My eigenvalues are λ=-1 and λ=-5. My basis are [1/3 1] and [1/2 1]. Now I have...
  10. karush

    MHB Understanding Initial Value Problems: Solving for y in y' = y-5 with y(0) = y0

    $\begin{align*}\displaystyle y'&=y-5\quad y(0)=y_0\tag{given}\\ y'-y&=-5\\ u(x)&=\exp\int-1\, dx = e^{-t}\\ (e^{-t}y)&=-5e^{-t}\\ e^{-t}y&=-5\int e^{-t} dt = -5e^{-t}+c\\ &y=-5\frac{e^{-t}}{e^{-t}}+\frac{c}{e^{-t}}\\ y&=\color{red}{5+(y_0-5)e^t}...
  11. karush

    MHB -b.1.2.2c initial value problem

    $\displaystyle \frac{dy}{dt}=2y-5, \quad y(0)=y_0$ rewrite $$y'-2y=-5$$ obtain u(x) $$u(x)=\exp\int-2\, dx = e^{-2t}$$ then $$(e^{-2t}y')=5e^{-2t}$$ just reviewing but kinda ?
  12. karush

    MHB De1.2.1 Solve the following initial value problem

    $\tiny{de1.2.1}$ $\textsf{ Solve each of the following initial value problems and plot the solutions for several values of $y_0$.}\\$ $\textsf{ Then describe in a few words how the solutions resemble, and differ from, each other.}\\$ $$\begin{align*}\displaystyle \frac{dy}{dt}&=-y+5...
  13. evinda

    MHB Initial value problem of the wave equation

    Hello! (Wave) I want to prove that if for the initial value problem of the wave equation $$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$ the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support. I have...
  14. karush

    MHB 2.1.7 Find the solution of the given initial value problem.

    $\tiny{2.1.{7}}$ $$\displaystyle y^\prime +y =\frac{1}{1+x^2}, \quad y(0)=0$$ $\textit{Find the solution of the given initial value problem.}$ \begin{align*}\displaystyle u(x) &=e^x\\ (e^x y)'&=\frac{e^x}{1+x^2} \\ e^x y&=\int \frac{e^x}{1+x^2}\, dx\\ %\textit{book answer} &=\color{red}...
  15. evinda

    MHB Solving the Initial Value Problem for a Wave Using the Forward Euler Method

    Hello! (Wave) We consider the initial value problem $$x'(t)=-y(t), t \in [0,1] \\ y'(t)=x(t), t \in [0,1] \\ x(0)=1, y(0)=0$$ I want to solve approximately the above problem using the forward Euler method in uniform partition of 100 and 200 points. I have written the following code in...
  16. yecko

    Initial value problem question

    Homework Statement Homework Equations ## y(t)\mu(t) - y(t_0) \mu(t_0) = \int_{t_0}^t \mu(s) g(s) ds## ## y(t) = \frac{1}{\mu(t)} \left[y_0 \mu(t_0) + \int_{t_0}^t \mu(s)g(s) ds\right]## The Attempt at a Solution (7 lines)I have done the first part, which seems correct, yet I am stuck with...
  17. karush

    MHB Solve the initial value problem

    Solve the initial value problem for $y$ as a function of $x$ \begin{align*}\displaystyle \sqrt{16-x^2} \, \frac{dy}{dx}&=1, \, x<4, y(0)=12 \end{align*} assume the first thing to do is $\int$ both sides
  18. M

    Solution to complex valued ODE

    Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...
  19. Drakkith

    Differential Equation Initial Value Problem

    Homework Statement A Solve the following initial value problem: ##\frac{dx}{dt}=-x(1-x)## ##x(0)=\frac{3}{2}## B. At what finite time does ##x→∞## Homework EquationsThe Attempt at a Solution ##\frac{dx}{dt}=x(x-1)## ##\frac{dx}{x(x-1)}=dt## Partial fractions...
  20. Drakkith

    Solving an Initial Value Problem for x(t=ln2): Step-by-Step Solution

    Homework Statement Solve the initial value problem: ##\frac{dx}{dt} = x(2-x)##, ##x(0) = 1## for ##x(t=ln2)##. Homework EquationsThe Attempt at a Solution I moved the right side to the left and multiplied both sides by dt to get: ##\frac{dx}{x(2-x)} = dt## Integrating gave me...
  21. Vanessa Avila

    Initial Value Problem for (DE)

    Homework Statement dv/dt = 9.8 - (v/5) , v(0) = 0 (a) The time it must elapse for the objet to reach 98% of its limiting velocity (b) How far does the object fall in the time found in part (a)? Homework Equations (dv/dt)/(9.8-(v/5)) The Attempt at a Solution I'm a little overwhelmed by this...
  22. Cocoleia

    Non exact differential equation, initial value problem

    Homework Statement I am trying to solve the following: y'''-9y'=54x-9-20e^2x with y(0)=8, y'(0)=5, y''(0)=38 Homework EquationsThe Attempt at a Solution The right answer is: y= 2+2e^3x+2e^(-3x)-3x^2+x+2e^2x I am only wrong on the coefficients C2 and C3. Where did I mess up in my solution?
  23. Cocoleia

    Initial value problem - differential equations

    Homework Statement I am given (y^2 + y sin x cos y) dx + (xy + y cos x sin y) dy = 0, y(0) = π/2 . I need to solve this Homework EquationsThe Attempt at a Solution At this point they still aren't exact, so I gave up. I can't figure out what the problem is. Is it possible that I have to...
  24. Kanashii

    4th order RK to solve 2nd order ODE

    Homework Statement Consider the initial value problem x" + x′ t+ 3x = t; x(0) = 1, x′(0) = 2 Convert this problem to a system of two first order equations and determine approximate values of the solution at t=0.5 and t=1.0 using the 4th Order Runge-Kutta Method with h=0.1. Homework Equations...
  25. karush

    MHB 242.14.2. solve the initial value problem

    $\tiny{242.14.2}\\$ $\textsf{(a) Verify that y = $Cx^2+1$ is a general solution to the differential equation $\displaystyle x \frac{dy}{dx}=2y-2$}$ $\textsf{(b) Use part (a) to solve the initial value problem $\displaystyle x \frac{dy}{dx}=2y-2, \, y(2)=3$}$ $\textit{all new so kinda ??}$
  26. Telemachus

    Initial value problem, finite differences

    Homework Statement Given an initial value problem: ##x'(t)=f(t,x)\,,x(t_0)=x_0## Use centered finite differences to approximate the derivative, and deduce a scheme that allows to solve the (ivp) problem. Homework Equations For centered finite differences ##\displaystyle\frac{dx}{dt} \approx...
  27. A

    Solving Initial-value Problem: Different "C" Values Explained

    Wondering why getting different values of "C" depending on how I solve the question. Not sure the values are different. Thanks. 1. Homework Statement Solve the initial value problem cos(x)Ln(y) \frac{dy} {dx} =ysin(x) , y>0, y(0)=e2. Homework Equations N/A. The Attempt at a Solution ∫...
  28. King_Silver

    Initial Value Problem (complex example)

    I know the method and can solve other initial value problems. This is the question given: dy/dx + y(-2) Sin(3x) = 0 for t > 0, with y(0) = 2. I've brought the dy/dx and let it equal to the rest of the expression so it is now: dy/dx = -y-2 Sin(3x) , with y(0) = 2 (i.e. when x = 0, y = 2 ) The...
  29. T

    Linear Differential Equation - Initial Value Problem

    Hello, I'm struggling with a simple problem here. It asks me to solve the following initial value problem: So far I've calculated the integration factor μ(x) = ex-x2 and I multiplied both sides of the equation by it and got this...
  30. Aristotle

    Initial Value Problem using Laplace Transform help?

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..) y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF -...
  31. S

    Solve Initial Value Problem and Determining Interval

    Homework Statement Solve the initial value problem and determine at least approximately where the solution is valid (2x-y) + (2y-x)y' = 0 y(1) = 3 Homework EquationsThe Attempt at a Solution I know how to solve it, and I got the correct answer, which was: 7 = x^2 - yx + y^2 and then applying...
  32. S

    Engineering Initial and final values for second order circuits

    Homework Statement I am attempting to understand this example shown below: Homework Equations During stead state DC, the capacitor is an open circuit and the inductor is short circuited. The Attempt at a Solution [/B] The questions I have are really related to the concepts as I don't...
  33. B

    Ζ in a intial value problem?

    hi, if there is a initial value problem with a ζ in it with specified values what do you do with it when taking the laplace transform?
  34. M

    MHB Initial Value Problem for a System of Linear Differential Equations

    Hey! :o We have the following initial value problem: $$x' = \frac{1}{2}(45 − x) + \frac{1}{4}(y − x) \\ y' = \frac{1}{4}(x − y) + \frac{1}{2}(35 − y) + \frac{1}{2}(z − y) + 20 \\ z' = \frac{1}{2}(y − z) + \frac{1}{2}(35 − z)$$ This can be written as follows: $$\begin{pmatrix} x\\ y\\...
  35. evinda

    MHB Solving Initial Value Problem: Determine Solution $y$

    Hi! (Smile) Consider the initial value problem $$\left\{\begin{matrix} y'(t)=\sqrt{|y|}, 0 \leq t \leq 2\\ y(0)=1 \end{matrix}\right. \tag 1$$ Show that for this problem the assumptions of the following theorem hold: "Let $c>0$ and $f \in C([a,b] \times [y_0-c, y_0+c])$. If $f$ satisfies at...
  36. J

    Solve Wave Equation: e^(-x^2), x*e^(-x^2), -infinity<x<infinity

    Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...
  37. D

    Differential equations - backwards problem

    Homework Statement If y=y(t) is the solution of the initial value problem { y'+(2t+1)y=2cos(t) y(0)=2 then y''(0)=? it is a multiple choice practice problem with choices y''(0)=2 y''(0)=-2 y''(0)=4 y''(0)=0 y''(0)=-4Homework EquationsThe Attempt at a Solution Im really not sure how to go...
  38. K

    Solving a First Order Initial Value Problem

    Homework Statement If y = y(t) is the solution of the initial value problem y' + (2 t + 1) y = 2 cos(t) y(0) = 2 What is y''(0)? Homework EquationsThe Attempt at a Solution Since this is a first order linear, I started out by finding the integrating factor so I can find what y is, and then...
  39. A

    Where is a solution valid in an initial value problem?

    The problem is from Adam's Calculus (7th Ed). It is an initial value problem, and I solved it: \begin{cases} y'=\frac{3+2x^{2}}{x^{2}} \\ y(-2)=1 \end{cases} \\ \implies y=-\frac{3}{x}+2x+\frac{7}{2} I can see that the solution is not valid for x=0, but the book says that the solutions is...
  40. D

    Solving this ODE for an initial value problem

    Homework Statement x \frac{du}{dx} \ = \ (u-x)^3 + u solve for u(x) and use u(1) \ = \ 10 to solve for u without a constant. Homework Equations The given hint is to let v=u-x The Attempt at a Solution This equation is not separable and the book wants me to make it separable...
  41. J

    Solving an Initial Value Problem with a Sinusoidal Differential Equation

    For the following problem \frac{dw}{d\theta}=\theta w^{2}sin(\theta^{2}), w(0)=1 I am not able to obtain the solution w=\frac{2}{1+cos(\theta^{2})} Can anyone point out my mistake? I have attached my working out in a picture format below (may need to enlarge it) thanks
  42. J

    Solving an Initial Value Problem with Separable Differential Equations

    Homework Statement The problem is from Walter Gautschi - Numerical Analysis, exercise 5.1. Consider the initial value problem \frac{dy}{dx}=\kappa(y+y^3), 0\leq x\leq1; y(0)=s where \kappa > 0 (in fact, \kappa >> 1) and s > 0. Under what conditions on s does the solution y(x) =...
  43. H

    Differential Equation Initial Value Problem

    Homework Statement I managed to work this problem all the way through, but I am in no way certain of my answer. I'd greatly appreciate any insight! Find the solution of the initial value problem. y'''+4y'=x, y(0)=y'(0)=0, y''(0)=1 Homework Equations Just for clarification...
  44. O

    MHB Uniqueness of Solution for x' = f(x) = √(1-x^2), x(2) = 1

    given this equation x' = f(x)= square root(1-x^2) x(2) = 1 I hae to show that teh solution is not unique my work: i tried to find the interval in which f(x) is defined, i said: 1-x^2 ≥ 0 (because of the sqrt) -x^2 ≥ -1 x^2≤ 1 x≤ ±1 my problem is if i take a number < 1 and substitute it on f(x)...
  45. S

    MHB Solving the Initial Value Problem for x'=x^3 with x(0)=1: A Step-by-Step Guide

    solve the initial value problem: x'=x^3 x(1)=1 my work dx/x^3 =dt then I integrated wrt t and obtained x^(-2) = t + c(c0nstant) where then this is 1/x^2 =t+c 1/x = square root of (t+c) then x= 1/sqrt(t+c) now when i apply the Initial value problem i get c = 0 and that is incorrect. where am...
  46. M

    Homogeneous initial value problem

    Homework Statement 4y" + 4y' + 5y = 0 y(0) = 3 y'(0) = 1 Homework Equations yh = e^ax(c1cosbx + c2sinbx) The Attempt at a Solution For the roots I got -1/2 + i and -1/2 - i so my a = -1/2 and b = 1 then I have to differentiate yh = e^(-1/2x)[c1cosx + c2sinx] this is where I get this...
  47. J

    Initial Value problem 1st order ODE

    Homework Statement I have been trying to follow a solution to a problem I had but do not quite understand the whole thing. I wondered if anybody could clear it up for me. Let a_0 be the initial value of 'a' for which the transition from one type of behaviour to another occurs. The...
  48. A

    Laplace transform initial value problem-

    Laplace transform initial value problem--need help! Looking at the solutions to these initial value problems, I am very confused as to how the highlighted steps are derived (both use heaviside step functions). I know the goal is to get the fractions in a familiar form so that one can look them...
  49. J

    MHB Integrating factor, initial value problem

    $ kxy \frac{dy}{dx} = y^2 - x^2 \quad , \quad y(1) = 0 $ My professor suggests substituting P in for y^2, such that: $ P = y^2 dP = 2y dy $ I am proceeding with an integrating factor method, but unable to use it to separate the variables, may be coming up with the wrong integrating factor ( x )
  50. F

    MHB Initial value problem for exact equations

    I've got a few small questions I'd like to straighten out. I'm really trying to establish a firm procedure involving the steps I write down because I find it helps me learn the math and avoid errors. Solve the initial value problem: (x+y)^2 dx +(2xy+x^2-1)dy = 0 with y(1)=1 So let M(x, y)...
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