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. 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. 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. 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. 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. 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...

11. 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 ?

35. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. MHB Solving the Initial Value Problem for x'=x^3 with x(0)=1

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. 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. 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. 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. 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. 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)...