Initial value problem Definition and 178 Threads

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...

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...

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

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...

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...

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!

$\tiny{31.6}$ 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...

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...

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...

$\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}...

$\displaystyle \frac{dy}{dt}=2y-5, \quad y(0)=y_0$ 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 ?

$\tiny{de1.2.1}$ $\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...

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...

$\tiny{2.1.{7}}$ $$\displaystyle y^\prime +y =\frac{1}{1+x^2}, \quad y(0)=0$$ $\textit{Find the solution of the given initial value problem.}$ \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}...

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...

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...

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

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...

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...

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...

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...

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?

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...

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...

$\tiny{242.14.2}\\$ $\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 $\displaystyle x \frac{dy}{dx}=2y-2, \, y(2)=3$}$ $\textit{all new so kinda ??}$

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...

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 ∫...

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...

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...

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 -...

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...

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...

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?

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\\...

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...

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...

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...

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...

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...

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...

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

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) =...

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...

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)...

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...

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...

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...

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...

$ 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 )

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)...