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.
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...
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...
$\tiny{de1.2.1}$
$\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$$
the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support.
I have...
Hello! (Wave)
We consider the initial value problem
$$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! (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)...