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
45 views

Solving this ODE 1

Trouble solving this ODE : $$\frac{d^2y}{dx^2}=\int_{-\infty}^{x^2/2} e^{x-t^2/2} \, \mathrm{d}t$$ $$x>0,\, y(0)=0,\, \frac{dy}{dx}(0)=0$$ in the form $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
0
votes
0answers
28 views

Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
2
votes
0answers
15 views

General Reaction Diffusion Equation Cranck Nicolson

How can i write matlab code that general reaction-diffusion equation by using Crank Nicolson Method? ∂u(x,t)/∂t= D(u(x,t))+R(u(x,t)) where D and R spatial discretizations (matrices) of linear or ...
3
votes
1answer
64 views

Solving $ \frac {dy}{dx} = \sqrt{y} x\cos(x) $ with $y(0) = 1$

I was helping someone work this problem out for an online course and I thought it'd be pretty easy since it's a first order separable DE. I ended up with $$ y = \frac {(x\sin(x) + \cos(x) + 1)^2}{4} ...
0
votes
1answer
20 views

finding solution to $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$

I tried to solve this equation with the coordinate method but I got a bit different answer compared to the suggested one by the solution manual. Where I am making the mistake in my solution? My ...
1
vote
0answers
23 views

finding solution to $au_x+bu_y=f(x,y)$ where $a \neq 0$

I can't get the solution in the required form of $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$$ "where g is an arbitrary function of one variable, L is the characteristic line segment from ...
1
vote
1answer
37 views

Show that if $T(x)=\int_0^\infty e^{-t}t^{x-1}dt $, then $T(1)=1$

I need to calculate the following Given: $ T(x)=\int_0^\infty e^{-t}t^{x-1}dt $ I need to show that $ T(1)=1 $ Solution: My logic was to plug $1$ in for $x$ before I integrated, but I am not ...
-1
votes
0answers
29 views

Definition of Laplace transform

I need to use the definition of laplace transform to determine $L(s)$ where $f(t)=e^{-t}$ on $0\leq t\leq 3$, and $2$ on $t>3$. My solution $$\begin{aligned} \int_0^3 e^{-st} e^{-t} dt + \int_3^{...
-2
votes
0answers
32 views

Linear Differential Equations help [closed]

$(y+x^3+x y^2)dx-dy = 0$ linear differential equation. help ASAP
-1
votes
1answer
34 views

Why is $y=h(x)$ deemed as a solution to ODE [closed]

Can some one explain why does $y=h(x)$ deemed as a solution to a given ODE concept? it seems like an important concept, and simple too. But I must be missing some peace in this logic. Can some one ...
1
vote
0answers
17 views

What are the methods for solving ODEs with accuracy higher than Runge Kutta 4?

Usually, justification of using RK4 is the following: "RK4 demonstrates a better approximations than Euler and Modified Euler methods of solving ODEs and offers a good balance between accuracy and ...
0
votes
0answers
36 views

How to pass from an ODEs system to reactions?

I have the following system: $$\frac{dx}{dt}=a_1+\frac{b_1x^n}{K_1^n+x^n}-gxy-d_1x,$$ $$\frac{dy}{dt}=a_2+\frac{b_2x^m}{K_2^m+x^m}-d_2y,$$ where $a_1,a_2,b_1,b_2,K_1,K_2,g,d_1,d_2,n,m$ are real ...
3
votes
2answers
98 views

Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
1
vote
1answer
49 views

How Can solve a n order Differential Equations

How can I solve the following equetion? what is the $$h(z).$$. $$z^n (z^n+1).|h'(z)|^n=const.$$.
0
votes
0answers
16 views

Help with linearization using Taylor Series

If I sound rather clueless, it is because I am. I'm having trouble with linearizing the following non-linear system: $$ 2\frac {dy(t)} {dt} = -y(t) - 0.9u(t)³ + 1.4q(t) $$ Where u(t), q(t) are ...
1
vote
1answer
26 views

How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
0
votes
2answers
46 views

equation $u_x+u=e^{2y+x}$ (part of the solution to $u_x+u_y+u=e^{x+2y}$)

I solved/analyzed the below PDE $$\left\{\begin{matrix} u_x+u_y+u=e^{x+2y}\\ u(x,0)=0 \end{matrix}\right.$$ and have a question to the one of the steps involving the integration, see below ...
-1
votes
2answers
56 views

Calculate an inverse Laplace transform

I need to calculate the inverse Laplace transform of $$\frac{s-2}{(s+1)^4}$$ Not quite sure how to do this one. I see that you should break the numerator up into $$\frac{s}{(s+1)^4}-\frac{2}{(s+1)^4}$...
0
votes
1answer
21 views

Integrating $u^2(x+y)$ over $\partial B_1 (0)$, where $u$ is harmonic

Let $u : \mathbb{R}^n \longrightarrow \mathbb{R}$ be a harmonic function. I must prove that $$\int_{\partial B_1 (0)} u(x+ty) u \left( x+\frac{1}{t}y \right) dS_y = \int_{\partial B_1 (0)} u^2(x+y) ...
0
votes
0answers
12 views

What does it mean to project all rotations of a function onto a different function?

Steerable filters allow me to optimally select an orientation of a filter without iterating over the set to find the optimal response (I'm doing this with a computer). The exact words from the paper ...
0
votes
3answers
53 views

Mapping a PDF to a uniform distribution on $(0,1)$

Let me preface this by saying that I'm not familiar with differential equations, other than basic "separable" differential equations. This problem has come up in a Probability problem that I am doing. ...
1
vote
0answers
17 views

Computational Methods for Determining Stability

For a nonlinear system $\dot{x} =f(x,\alpha)$ where $\alpha$ is a parameter, with fixed points $x^*$ such that $f(x^*,\alpha) = 0$, what methods are there for computationally determining the stability ...
2
votes
1answer
50 views

Modified Wave Equation: Bound $\int u^2 \, dx$

I'm studying for a qualifying exam and I can't figure this problem out: Suppose $B \subset \mathbb R^n$ is the unit ball centered at the origin and that $u$ is a smooth solution of \begin{align*} ...
1
vote
3answers
58 views

Separation of Variables and Linear PDEs

Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method ...
0
votes
0answers
7 views

schwarz decomposition methods on shallow water equations [closed]

How can I apply schwarz decomposition methods on two dimensional shallow water equations mathematically. Thanks
1
vote
1answer
42 views

Could you give me the proof of this ODE problem?

Let x : [0,∞) → $R^3$ denote a solution of the ODE x'(t) = c$\times$x(t), $x(0)$ = $x_0$. Here c is a fixed unit vector, × is the standard cross product on $R^3$ . Show that the trajectory of x lies ...
1
vote
0answers
64 views

Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
0
votes
0answers
28 views

Solution of a system of two differential equations

I have the following system of two differential equations: \begin{align} \frac{\mathrm{d}^2E_x(z)}{\mathrm{d}z^2}-\alpha^2 E_x(z)+c E_y(z)=0,\\ \frac{\mathrm{d}^2E_y(z)}{\mathrm{d}z^2}-\beta^2 E_y(z)+...
0
votes
0answers
16 views

Computationally Determine Dynamics on a Center Manifold

I have a four dimensional system of nonlinear differential equations $\dot{x} = f(x)$ with a single parameter $\alpha$ and Jacobian matrix $J$ with eigenvalues $\lambda_1 = 0, Re(\lambda_2) < 0,Re(\...
1
vote
3answers
32 views

Inhomogenous Differential system

This is the system, $$ y'_1 =y_1+y_2+1 $$ $$ y'_2= -y_1+y_2+1 $$ initial value problem which fulfill: $$y_1(0)=1$$ $$y_2(0)=-1$$ value to find $$y_1(π)= \text{?}$$
1
vote
1answer
32 views

Why $f_y(x+\delta x,y)=f_y(x,y)+o(1)$?

I am currently doing some independent study and came across a set of notes, within these notes a proof of the multivariable chain rule is given. The proof it gives is: Consider an arbitrary ...
0
votes
1answer
49 views

Why are Lyapunov functions always quadratic?

Consider stable linear system $\dot x= Ax + Bu$. We’ll show that the Lyapunov bound is tight with $V (z) = z^T W^{−1}z$. Multiply $AW_c + W_c A^T + B B^T = 0$ on left & right by ${W_c}^{−1}$ to ...
1
vote
2answers
81 views

Nonlinear 2nd order ODE

I have been looking at numerical solutions to the following nonlinear Bessel-type ODE: $$ xy'' + 2 y' = y^2 - k^2, $$ where k is a constant. In general, $y = \pm k$ is an asymptotic solution, and as $...
0
votes
0answers
28 views

Why solution vectors of an ODE form a basis of the solution space?

I am currently doing some independent study in preparation for university in October and stumbled upon some online notes. Within these notes it states: "Let $y_1$ and $y_2$ be linearly independent ...
0
votes
1answer
23 views

Special solution of following system of differential equations

Suppose now system of differential equations, namely, $$ \begin{equation} \ddot{y}(t) + \omega^{2}y(t) = \dot{z}(t) \\ \dot{z}(t) = (-A+\dot{y}(t))z(t)\end{equation} $$ I want to check, for which $y(t)...
4
votes
0answers
37 views

Clarification on asymptotically stability of dynamical systems

I'm wondering if someone can provide a clarification between 2 seemingly opposing definitions from reputable sources on dynamical systems! My Russian textbook, "Dynamical Systems I: Ordinary ...
-2
votes
1answer
25 views

Partial differential equation solvable as ordinary differential equation [closed]

Solve the following partial differential equation for $u=u(x, y)$: $$U_{yy}+6U_y+13U= 4e^{3y}.$$
1
vote
0answers
19 views

Converting system of m rational ODEs into one n-order rational ODE preserving regularity

Who knows anything about the Conjecture, which claims that for any explicit system of m 1st order rational ODEs x'(t) = Rational RHS y'(t) = Rational RHS z'(t) = Rational RHS . . . . . . ...
0
votes
0answers
15 views

Method of characteristics for Euler equations (expansion wave)

The Riemann invariants along the characteristic curves (eigenvalues) for Euler's equations in the isentropic case (e.g. expansion wave) can be written as: $\phi_+ = u + \frac{2a}{\lambda - 1}=const $ ...
2
votes
6answers
109 views

How to solve $\frac{\mathrm dy}{\mathrm dx}=\frac{x+y}{x}$

I am looking for a help with the below equation $$\frac{\mathrm dy}{\mathrm dx}=\frac{x+y}{x}$$ I don't get the suggested answer $$y = x\ln(x) + c x$$ My attempt $$\mathrm dx(1+\frac{y}{x})=\mathrm ...
0
votes
0answers
25 views

Constant solutions to the Lotka-Volterra equations

Let $$x' = ax - bxy, \\ y' = -cy + dxy,$$ be the Lotka-Volterra equations with $a,b,c,d > 0$. How to show that for every choice of $a,b,c,d$ there is an initial condition $x(0) = x_0$ and $y(0) = ...
0
votes
1answer
21 views

Existence and uniqueness of an ODE solution across simple discontinuities

I am studying differential equations using MIT's publicly available materials. One of the exercises runs as follows: Let $I = (a,b)$ be an open interval containing $0$, and consider the ODE \begin{...
4
votes
1answer
26 views

ODE system solving by sequence of functions

Let $y' = Ay$ where $A = \begin{pmatrix} 0&1 \\ -1& 0 \end{pmatrix}$ and $y( 0 ) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Consider the map $$G: C(\mathbb{R},\mathbb{R}^2) \to C(\mathbb{R},\...
0
votes
0answers
11 views

System of nonlinear first order pde's

I need a reference to results on existence of solutions to systems of nonlinear first order PDEs. to be more precise I am interested in the following: Let $F\colon\mathbb{R}\to \mathbb{R}^d$ be a ...
1
vote
0answers
66 views

What is wrong with my formulas for a mathematical model of a double pendulum?

I wanted to create a computer simulation on Matlab, using a model for a pendulum from this study (A double pendulum model of tennis strokes. Rod Cross. Uni of Sydney, 2006) - Link I wanted to use the ...
0
votes
2answers
47 views

laplace differential equation with conditions

I have to solve this differential equation with laplace $y'' + 6y' + 9y = \begin{cases}5t & 0 < t \le 3 \\ 0 & t>= 3\end{cases}$ and $y(0)=1, y'(0)=1$ I know what to do with the left ...
2
votes
2answers
38 views

Weak problem formulation for PDE and boundary conditions

Consider the following example: $$ - \Delta u = f \mbox{ in } \Omega, $$ $$ u = 0 \mbox{ on } \Gamma, $$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
0
votes
0answers
31 views

Solution to wave equation for $u(t, 0) = \gamma(t)$

I am looking for a necessary and sufficient condition on a function $\gamma: \mathbb{R} \to \mathbb{R}$ such that there is a solution $u \in C(\mathbb{R}^2)$ for the wave equation $$\partial^2_t u(t, ...
1
vote
1answer
24 views

Solving ODE by substitution. Where does $dy$ goes

When solving ODE by substitution, where does $dy$ goes from the following example? $$\left(1+\frac{sin(y)}{cos(y)}\right)dy=x dx$$ Let $u=-cos(y)$. Hence $du = sin(y)$, which results in the following: ...
2
votes
1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...