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

learn more… | top users | synonyms (1)

0
votes
0answers
3 views

Classifying boundary conditions when PDE is given on the whole space.

I am being asked to classify the boundary value problem: $u_{x} + u_{y} = 0$ such that $u(x,y)=1$ whenever $x=y$. I have only learned about three different boundary conditions: Dirichlet, Neumann, and ...
0
votes
0answers
4 views

Differential Equations SEIR Model

Assume an SEIR model is appropriate for a particular disease. Write down the differential equations for the model, with the assumption that the parameter names are beta (transmission rate), kappa ...
1
vote
0answers
21 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$ such that $f(x_0)> 0 $. $x(t)$ is the solution to $x'=f(x)$ such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all $t \geq 0$ if and ...
0
votes
0answers
34 views

Hairy ball Theorem: ODE/PDE applications?

Hairy Ball Theorem: There is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. I do know that it was Poincaré who conjectured that result in 1881 when he studied ...
0
votes
0answers
16 views

Show that IVP has solution

Show that the initial value problem $$y''+by'+cy=g(t)$$ with $y(t_o)=0=y'(t_o)$, where $b,c$ are constants has the form $$y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,$$ one method is to separately consider ...
2
votes
1answer
18 views

Separation of variables - “formal” notation?

When using separation of variables technique to solve differential equations, I sometimes have both f(x) and g(y) on the right side, and then I divide by g(y) to separate them.... but how can I ...
1
vote
0answers
56 views

Solving differential equation.

In my research work I need to find the solution of the following differential equation. $\displaystyle y'(x)=\frac{y(x)+1}{2 \sqrt{x y(x)}-x},$ $y(0)=0$, where the solution must satisfies the ...
0
votes
1answer
22 views

Am I wrong with this simple differential equation?

I have been given the following differential equation: $$(x^2 + y)dx - xdy = 0.$$ The equation turns out to be inexact, so I opted for the simple straightforward solution with the integrating factor ...
0
votes
2answers
27 views

Find the general solution to differential equation

Could someone please help me with this if $\frac{dx}{dt} = 2x$ and $x(1)=x(0)+1$ find $x(t)$. I started off with: $$\frac12 \int \frac1x \, dx = \int 1\, dt$$ $$\frac12 \ln(x) = t+c$$ ...
1
vote
0answers
31 views

Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
0
votes
1answer
22 views

Existence and uniqueness of an ODE

I have the following ODE: $y'=y+e^{-2t}y^2$ I know the solution is $y=\frac{1}{e^{-2t}+ce^{-t}}$, c constant. Then the problem says that $y=0$ is a different solution. How can I explain that this ...
2
votes
1answer
22 views

Rewrite a differential equation formula

I am currently reading Elementary Differential Equations and I don't quite understand how they rewrote this Differential equation. I know that it is a very simple answer, but for some reason I can't ...
0
votes
0answers
16 views

The difference between euler line and direction field?

Can someone please explain to me what the difference is between the direction field and euler lines? Direction field is the tangent line to the integral curve of a differential equation, as far as I ...
0
votes
0answers
17 views

Finding solutions for a nonlinear ODE of the second order

Try to solve in closed form the ODE (any kind of special functions can be used, or a solution in parametric form is also valid) yy'' + y' = x, (1) y(x_0) and y'(x_0) are open for any x_0. Note ...
2
votes
1answer
95 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$ such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent while $\,f^{(n)}=f$. Could you give me some hints? I truly ...
0
votes
0answers
16 views

Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
2
votes
2answers
25 views

If $f(x) \cdot x < 0$ for all $x \in \partial B_R(0)$, then the IVP $x' = f(x)$, $x(0) = x_0$ has a global solution.

I have a homework problem that asks If $f : \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and satisfies $$ f(x) \cdot x < 0 \quad \quad \text{for all } x \in \partial B_R(0) ...
1
vote
1answer
19 views

Generalizing Method of Characteristics: Example

I have consulted other sources, but the relevant ones have used notation that I am not entirely comfortable with. I'm told that generalizing this procedure is straightforward, but I find myself ...
7
votes
1answer
116 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
1
vote
3answers
33 views

Differential equation using substitution

I have the following differential equation with a hint to solve of using the substitution of $u = \frac{y}{x}$: $$\frac{dy}{dx} = \frac{(y/x)^3 + 1}{(y/x)^2}$$ I was wondering, after the ...
0
votes
2answers
37 views

Solutions of a second order nonlinear differential equation

I am in trouble solving the following differential equation: $$\ddot{x}(t)=-\alpha\dot{x}(t)\dfrac{x(t)}{\left(\beta^2+x(t)^2\right)^{\frac{5}{2}}}$$ where $\alpha$ and $\beta$ are constant. How can I ...
0
votes
2answers
37 views

First order differential equation with time-varying parameter

If I have: $\dot{\sigma}(t) = -\gamma \sigma(t)$ where $\gamma$ is a constant, the solution is given by: $\sigma(t) = \sigma(0) e^{-\gamma t}$ Now what if I have the differential equation: ...
0
votes
0answers
24 views

find second order differential equation

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ My question is how to find $f$? Can show some steps? Thanks for help.
0
votes
1answer
20 views

seperation of variables difficult integral

$x'(t)=x(t)*(a-x(t)), x(0)=x_0$ I have to solve this ODE with seperation of variables. But I have problems. $\frac{\partial x}{\partial t}=x(t)*(a-x(t)) \iff ...
0
votes
0answers
31 views

Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature ...
0
votes
3answers
34 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
-1
votes
0answers
24 views

Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
1
vote
0answers
11 views

uniqueness of complete integral [on hold]

Is the first order nonlinear PDE that the given complete integral solves unique? I had this question while solving the problem 3-3 of "Partial Differential Equations" written by Evans.
3
votes
2answers
62 views

2nd order nonlinear ODE question

I am looking for help to solve the following $F(x,y(x),y'(x),y''(x))=0$ equation: $$ xy''(x)-y'(x)-(x^2)y(x)y'(x)=0 $$ Very much appreciated.
0
votes
1answer
33 views

Proving this Differential Equation has one solution

Suppose q(x) is an n-th degree polynomial and consider the following DE: x(dy/dx) + y = q(x) Show that there is only one solution to this differential equation that is, itself, a polynomial. What is ...
1
vote
0answers
28 views

Explicit solution to a first order nonlinear ODE

Is there any explicit solution to the following ODE? $G'(z) =aG(z)+bG(z)^α-c$ $G(0) = d_0 $
5
votes
2answers
118 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, ...
1
vote
0answers
24 views

Asymptotic behaviour of $\varphi''(x)=F(\varphi(x))$

I'm concerned with the discussion of a ODE, especially the discussion of the solution. I've got the assumptions that there is the relation $\varphi''(x)=F(\varphi(x))$ for all $x$ on $\mathbb{R}$. ...
0
votes
1answer
17 views

Can I extend these ODE formulas to complex numbers?

In my calculus class, we recently covered first-order, linear ODEs. Specifically, we discussed the formula for the solution of one (and its derivation): $$y=\frac{1}{u(x)}\int Q(x)u(x)dx$$ where ...
0
votes
1answer
25 views

Differential equation with no nontrivial periodic solution

We are given $f=(f_1,f_2): \mathbb{R}^2 \rightarrow \mathbb{R}^2$, $C^1$ class with the property: $$(1) \ \ \ \forall_{(x,y)\in\mathbb{R}^2} \frac{\partial f_1}{\partial x}+\frac{\partial ...
1
vote
1answer
24 views

how to write a differential equation for a problem like this

I've got a problem and i should solve it using differential equation.I don't know how to write the equation and start. A person is trying to fill a bathtub with water. Water is flowing into the ...
0
votes
0answers
15 views

homogeneous BVP has at most one linearly independent solution

I am trying to understand following proof. I understand the set up however can't make the connection with the Picard Lindelöf Theorem. Can you please help me with this? Statement: The homogeneous ...
0
votes
0answers
29 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
0
votes
1answer
29 views

How to solve this two variable Bernoulli equation ODE?

I'm trying to solve this homework question but the two variables is throwing me off. Which one is my standard $t$? How do I handle both variables? I'm to solve this Bernoulli equation via substitution ...
0
votes
1answer
22 views

is this differentiable form exact?

Take $\omega^{2}$ a 2-exact form and $\omega^{3}$ a 3-closed form, the question is, can we have that $\omega^{2}\wedge\omega^{3}$ be exact? Thanks!
0
votes
1answer
24 views

Solution of the heat equation

Let $u:\mathbb{R}^n\times(0,+\infty)\to\mathbb{R}$ solves the following heat equation: $$u_t(x,t)-\triangle u=0,\quad (x,t)\in\mathbb{R}\times(0,+\infty)$$ (a) Show that for each ...
1
vote
0answers
27 views

Differential Equation Find general solution of y'' - y =cosh(x) using variation of parameters

Hello I am having some issues with the simplification of the DE, I am okay up on till $$y_p(x)=v_1(x)y_{p1}(x) + v_2(x)y_{p2}(x) $$ $$ \frac12 e^{-x}\left(\frac {-e^{2x}}4-\frac x2\right)+ \frac ...
0
votes
1answer
35 views

Prove that $\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth(\frac{\pi}{2})$

I want to prove that: $$\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth\left(\frac{\pi}{2}\right)$$ Using the fourier series: $$\phi(x) = \begin{cases}0 & \text{if ...
0
votes
2answers
38 views

How to determine behaviour of this derivative in the following differential equation?

Given the following differential equation $$\frac {dx}{dt} = ax + \cos(x)$$ for some $a \in \mathbb R$. I need to determine the shape of the direction field of $x(t)$ in the vertical axis and $t$ in ...
1
vote
1answer
73 views

Looking for a way to solve this differential equation.

Can somebody give me a hint how to try to solve the following differential equation: $ \ddot{r} - \frac{1}{r^3} = 1$ where $r = r(t)$ and $\ddot{r}$ is the second derivative. It is not homework btw. ...
0
votes
1answer
23 views

What can I think of the function $F$ that's being used for most(?) explicit first order ODEs?

Almost anything on this topic only deals with how to solve ODEs, but so far I couldn't find one single site defining this ominous $F$ that's being used so often, like in Wikipedia or in my script (not ...
0
votes
0answers
11 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
0
votes
0answers
10 views

Differential Algebraic eqn using Adomian Decomposition Method [on hold]

Refer to research paper http://www.gbspublisher.com/ijpamsv3/ijpamsv3n1_10.pdf, in example 1 i am unable to understand how author has calculated u1,0 =14xsinx+sinx-xcosx. Can somebody plz ...
3
votes
1answer
164 views

On definition of gamma function.

We all know that gamma function's definition is $$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$ and it is divergent for $x<0$. Yesterday, I was studying about Bessel ...
2
votes
3answers
63 views

The function $4x^3y/(x^4+y^2)$ fails the Lipschitz condition near the origin

I have to prove that Lipschitz condition is not satisfied for the function, $$ f(x) = \begin{cases} {4x^3y \over x^4 +y^2}, & \text{if $(x,y) \neq (0,0)$ } \\ 0, & \text{if $(x,y)=(0,0)$ } ...