Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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0
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1answer
13 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rsinϴ, r = \frac{x}{sinϴ} , \frac{dr}{dx} = \frac{1}{sinϴ} $$ $$\frac{dA(x)}{dx} = ...
1
vote
1answer
9 views

Differential equation corresponding to a linear system of differential equation.

Consider linear system of differential equations $$\frac{dx}{dt}=ax+by$$ $$\frac{dy}{dt}=cx+dy$$ my question is how to find the second order linear differential equation corresponding to above ...
4
votes
1answer
75 views
+50

An MCQ on Greens function

$$G(x,t) =\begin{cases} a+ b\log t & \text{if $0<x<t$ } \\[2ex] c+ d\log t & \text{if $t<x<1$ } \end{cases}$$ is a Greens function for $xy''+y'=0$ subject to $y$ being bounded as ...
0
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0answers
19 views

System of Ordinary differential equations.

Consider the system $$\frac{dx}{dt}=(1+x^{2})y$$ $$\frac{dy}{dt}=-(1+x^{2})x,t\in\mathbb{R.}$$ With initial condition $(x(0),y(0))=(a,b)$ ,Then the system has solution $1.$ Only if $(a,b)=(0,0)$ ...
0
votes
1answer
12 views

Solving solution given initial condition condition

Suppose we know that: $$u_t=ku_{xx},~~~~~~~~0<x<l,~~~t>0$$ and $$u(x,t)=\sum_{i=0}^\infty[C_n~cos(n\pi x/l) ~e^{-w_nkt}]$$ where $w_n=\frac{n\pi}{l} ~~~ for~~n=1,2,3,...$ What if the ...
0
votes
1answer
19 views

Converting higher-order ODE to first order ODE

given $y''' + 2y'' -5y' = 2y + 5y^3$ convert to a system of first order equations. My question is do we need to make substitutions for $y$ and $y^3$ or are we only concerned with the derivatives, if ...
1
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0answers
11 views

For a nondecreasing map, if $\xi(a) < \eta(a)$, then $\xi(t) < \eta(t)$ for all $t \in [a,b]$.

I am studying the following theorem from Morris Hirsch's second paper on systems of differential equations which are competitive or cooperative: Let $V \subset \mathbb{R}^n$ be on open set and ...
1
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0answers
19 views

Error of the Numerov Method

The Numerov method is an iterative algorithm for solving second order differential equations. A full derivation is here on the Wikipedia page: https://en.wikipedia.org/wiki/Numerov's_method. I am ...
1
vote
1answer
21 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.
2
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1answer
14 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
0
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2answers
25 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
0
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0answers
18 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
0
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3answers
21 views

Convolution of $te^{2t}$ and $\delta_1-\delta_2$?

I seek to find $f*g$ where $f=te^{2t}$ and $g=\delta_1-\delta_2$ and $\delta_a(t)= \displaystyle \lim_{\epsilon \to 0^+}d_{a,\epsilon}(t)$; i.e. $\delta$ is the Dirac Delta function. We have learned ...
0
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3answers
100 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
-3
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1answer
25 views

Find the general solution of the pde. [on hold]

Find the general solution of the equation for $v=v(x,y)$: $$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=2xy(x^2-y^2)$$
0
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1answer
27 views

Solving $xy'+y=x^{k}$

Find a solution to: $$xy'+y=x^{k}$$ Where $k>0$ and $f$ and $f'$ exist. I understand that we can take the Laplace of all of the terms and then find the inverse Laplace transform to get a ...
0
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1answer
47 views

Model for spread of infection, with vaccination

I'm trying to solve following problem: $N = 10^6$ ... number of people $ir = 8\% $ ... infection rate time unit - 1 day And when there are 3% of population infected, vaccination begins. Its effect ...
0
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0answers
22 views

The Picard-Lindelof Existence theorem.

The version of theorem I am using. Let $t_0 \in \mathbb{R},a>0,R>0,y_0 \in \mathbb{R}^n$ and $f:[t_0,t_0 + a] \times B_R(y_0) \rightarrow \mathbb{R}^n$ (jointly) continuous, $|f(t,y)| \leq ...
0
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1answer
45 views

Solve the system of differential equations

I plan on adding more into later just a bit stuck, researching it at the moment. Solve the system of differential equations $$\begin{bmatrix} x'\\y' \end{bmatrix} - \begin{bmatrix} -11&15\\ ...
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2answers
14 views

Coefficients of homogeneous second order ODE when initial conditions are both 0

I'm currently trying to solve this problem: Find the unique solution to the differential equation $y''- y'- 6y = 18x$ With initial conditions $y'(0) = 0 , y(0) = 0$ My current solution is $y = -3x + ...
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0answers
24 views

Find the general solution of this pde?

I have a final in my PDE Class and needs some help with a review problem. Find the general solution of the equation for $v=v(x,y)$: $$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial ...
0
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1answer
31 views

Does the substitution $u = \dfrac{y}{x}$ still be applicable for homogenous functions of degree $k$?

Suppose that I have the differential equation: $$\dfrac{dy}{dx} = f(x,y) $$ According to the book, if $f(x,y)$ is homogeneous of degree $0$ ($f(\lambda x,\lambda y) = f(x,y)$) then we can use the ...
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0answers
16 views

ODE PDE qualifiers [on hold]

What are good resources online to work on problems for Graduate level candidacy exam on ODE and PDE. For ODE's we used Grimshaw's and Perkov's book. If theres' any solution manuals do let me know. ...
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0answers
27 views

analyze stability of a system of first order differential equations of different types [closed]

I am working in a mathematical model and I need to analyze the stability of the system of differential equations that define the model, but I don't know how. Here I show a system with the same ...
3
votes
2answers
904 views

Spring Calculation - find mass

A spring with an -kg mass and a damping constant 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 newtons. If the spring is stretched 5 meters beyond its natural length ...
-2
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0answers
43 views

Solving a specific second order ODE [on hold]

I need some one can help me to solve the following equation : $$z_{tt}-z_{xx}-2z_t = \alpha(t,x)(z_x-z_{tx})$$ where $\alpha(t,x) = \frac{4\epsilon x(1+\epsilon t)}{(1+\epsilon t)^2 - \epsilon^2 ...
0
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2answers
13 views

For what points $(x_0,y_0)$ does this problem actually have a unique solution on some interval $|x-x_0| \le h$?

We're using Picard's Theorem, namely: Let $f(x,y)$ and $\frac{\partial f}{\partial y}$ be continuous functions of $x$ and $y$ on a closed rectangle $R$ with sides parallel to the axes. If ...
0
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2answers
29 views

Example of a dynamical system which has an $\omega$-limit which is a cylinder of closed orbits

I have been studying dynamical systems and have recently come accross the following theorem: Suppose $n=3$. Let $L$ be a compact limit set which contains no equilibrium. Then: $L$ is either a ...
0
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3answers
18 views

Need to overcome erroneous result when differentiating natural log of a fraction

I am trying to differentiate the following: $$ln(3x-8/6x+2)$$ my (incorrect) method is: let $$ln(x) = ln(u)$$ therefore when differentiating u.. $$ln(u) = 1/u$$ and diff of$$$$(3x-8/6x+2) = 3/6 = ...
3
votes
1answer
49 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
3
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0answers
33 views

Counter example for uniqueness of second order differential equation

I have a second order differential equation, \begin{eqnarray} \frac{d^2 y}{d x^2} = H(x) y \quad \quad \quad * \end{eqnarray} where, $H(x) = \frac{sech(x) sech(x)}{x + \ln(2 \cosh(x))}$ . Plot of ...
0
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1answer
23 views

How to simplify the D.E to bring it to the Linear form?

The D.E is : $$\frac{dy}{dx}=1-x(x-y)-x^3(y-x)^2$$ Here the only problem is how to simplify the equation into the standard linear differential equation. After the solution the equation can be solved ...
0
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1answer
38 views

Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
2
votes
2answers
40 views

If $\varphi$ is bounded above, increasing, and concave down, does $x\varphi'(x)$ go to zero? How fast?

Suppose $\varphi: [0,\infty)\rightarrow [0,1)$ is an increasing differentiable function ($C^\infty$ if you want) with $\varphi \rightarrow 1$ and $\varphi'>0, \varphi''<0$. My question is: Is ...
0
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0answers
3 views

DAE with (almost) linear constraints

I have a system of the form: $$\begin{align} \frac{d}{dt}x^i &= a^i_jx^j \notag\\ \sum_i x^i &= 1 \notag \end{align}$$ where $x^i = x^i\left(t\right)$, $t\in\mathbb{R}_+$ and the $a^i_j$ are ...
0
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3answers
53 views

Is this a vector field?

One of the example questions we've been given in lectures is to plot the vector field given by the first order equation $y' = y - y^{3}$. The solution we were given looks like this: However, I ...
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0answers
31 views

Write out the explicit Kolmogorov forward differential equation

Let $(X_t)$ be a continuous-time Markov process with two states, as shown below. Assume that there are two positive numbers $a$ and $b$ such that for all times $t\geq 0$ and $h>0$, $P(X_{t+h} = 2 ...
0
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0answers
8 views

What is the importance of Leray Schauder non linear alternative [on hold]

I'm questioning the importance of the Leray-Schaulder alternative, isn't more simple to use directly the Schauder fixed point theorem. Especially that, to prove the alternative we use the fixed ...
0
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3answers
48 views

Solving the IVP given by $\dot x=\frac{t-x}{t+x}$ and $x(0)=1$

Find all solutions for $\dot x=\frac{t-x}{t+x}$ with $x(0)=1$. I am seriously struggling to separate the variables since the fraction is quite complex. How may I be able to separate $x$ and $t$?
0
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1answer
36 views

Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

In the plane find two solutions of the initial-value problem $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$. I think we get to use the method of characteristics But I am not ...
0
votes
1answer
28 views

Solving PDE $v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$

We are given a PDE with $$v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$$ $$ v(T,x) = x^2 $$ for $0 < t \leq T$ and $x \in \mathbb{R}$ So far I have found that using the Feynmann-Kac equation, we get ...
1
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0answers
57 views

Cost Function Neural Network With Weight Derivation

Given this cost function of a single-layerd neural network with a sigmoid function: $$E_j = \frac{1}{2} \sum_{k=1}^{K}(\text{target}_{jk} - \text{observed}_{jk})^2 + a\sum_{i=1}^{I}w_{ij}^2$$ I ...
1
vote
1answer
47 views

Differential Equation involving Lambert W function

I was wondering whether there is an explicit solution to the following differential equation $$f'(x) = g'(t)\left(f(t)\left(\frac{a}{g(t)} -1 \right)-\frac{a}{g(t) \lambda}\left( 1+ ...
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0answers
15 views

Kolmogorov forward and Backwards equation interpretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
0
votes
1answer
11 views

Linear ODEs with a constant on the RHS

I'm trying to solve an equation of the form $a*\frac{dC}{dt} + bC = c - ce^{-t}$ I've found the homogeneous solution but finding it difficult to get the particular solution, could anyone help?
0
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0answers
16 views

Minimal and maximal problem

Cuboid with contlstant volume $V$ its base is a rectangle length $nX$, width $X$ The whole area of cuboid is $A$ and $A$ was minimal area Prove with differentiation that $A^3n=54 V^2(1+n)^2$
2
votes
1answer
17 views

Symmetry of Green's function on the general case

Let's consider the differential equation $$\nabla\cdot[p(\mathbf{r})\nabla u(\mathbf{r})]-s(\mathbf{r})u(\mathbf{r})=-f(\mathbf{r}).$$ I want to show that the Green's function is symmetric, so that ...
1
vote
0answers
17 views

How to prove the sufficient condition for a PDE has positive eigenvalues

Strauss' textbook says it is $\left[f(x)f'(x)\vphantom{\dfrac11}\right]_{x=a} ^{x=b}<0$, or $=0$. the assumption is $f''(x)=-\lambda f(x)$ for real $\lambda$. they told me to use Green's first ...
0
votes
2answers
32 views

General Solutions to differential Equations

What are the steps to finding the general solution of a differential equation? More specifically how do I solve: $$\frac{dy}{dt} = t^{2}y + e^t$$
-5
votes
1answer
62 views

The method to solve a basic ODE [on hold]

Anyones can help me to solve the following ODE: $$\frac{dx}{dt} = \frac{x-t}{x+t}$$ Thanks a lot