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

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1answer
18 views

Repelling or attracting spiral phase portrait in canonical basis

If matrix $\bf{A}$ of a system $\bf{x}'=\bf{A}x$ (*) has only complex eigenvalues and eigenvectors with non-zero real parts, and we make the substitution $\bf{y}'=\bf{B}x$ (**), where ...
1
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2answers
28 views

Adjoint of a differential operator

I am self-studying differential equations using MIT's publicly available materials. One of the recitation exercises runs as follows: Define an inner or dot product on $\mathcal{C}[a,b]$ by ...
0
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1answer
15 views

First order ODE with hidden dependencies

I'm trying to solve the ODE $G'+f'(x)[k-G]=0$ where $G=G(g)$ and $x=x(g,h)$. I would like to solve for $G(g)$, but I'm not sure how to go about it. Is there a method to do it? Thanks!
0
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1answer
18 views

Finding General Solution by Integrating Factor Method for a Diff E.Q.

Find the General Solution $y'-2y = t^2e^{2t}$ I am stuck after finding $y = e^2t ∫e^{-2t}t^2e^{2t} dt$ Perhaps this isn't even right, but i believe I go forward with substitution.
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2answers
51 views

Solve $(1+x^{2}) \frac{\partial{u}}{\partial{x}} + y\frac{\partial{u}}{\partial{y}} = 0$

I have the following equation: $$ (1+x^{2}) \frac{\partial{u}}{\partial{x}} + y\frac{\partial{u}}{\partial{y}} = 0. $$ Explain why we cannot deduce a solution by imposing $u(x,0)$. So far what I ...
3
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2answers
38 views

How to construct an exact but non separable differential equation?

So I can prove that all separable differential equations are exact, and I can intuitively figure out that not all exact differential equations are necessarily separable, but I'm having a hard time ...
0
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0answers
17 views

Determine rate of level build up in horizontal cylinder when given volumetric flow rate.

A horizontal cylinder with diameter $D$ and length $L$ is being filled up with liquid at a rate of $Q$. The height of liquid level inside the horizontal cylinder is $h$. The volume of liquid is ...
0
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1answer
29 views

Is $e^{-r/2}$ equivalent to $r^{-(l+1)}$ in the radial solution of Laplace equation?

When we solve the Laplace equation for Hydrogen Wave Equation at large r, we obtain the expression below to account for the behavior of the wave at very very large $r$ $$R=e^{-(r/2)}$$ At very small ...
1
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0answers
44 views

Nonlinear ODE differential equation

The following equation arises when minimizing certain non-equilibrium actions: $$\frac{\partial^2}{\partial t^2}y=(1+|y|^\alpha)f(t)$$ where $\alpha > 0$ and $f(t)$ is non-negative, smooth, and ...
0
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2answers
22 views

A general solution for a 2d pdf (ode)

I have the following 2 dimensional PDE: $$ \partial_{x_1}^2 u(x_1,x_2)+\frac{1}{x_1^2}\partial_{x_2}^2 u(x_1,x_2)+\frac{1}{x_1}\partial_{x_1}u(x_1,x_2)=k $$ where $k>0$ is a constant, and ...
0
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1answer
15 views

Homogenous ordinary differential equations

Find a family of solutions for the following equation, assume that the coeeficient of dy $\not= 0$ $xy' - y - x sin(\frac{y}{x}) = 0 $ The solution I get when I solve it using the substitution u = ...
0
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1answer
45 views

Solving the following differential equation: $\frac{dN}{dt} = rN(1-N^2)$.

Here, r is a positive constant and has initial condition N(0) = $N_0$, a constant. My attempt: $\frac{dN}{dt} = rN(1-N^2), \\ \frac{1}{N(1-N^2)} dN = r dt, \\ (\frac{1}{N} - \frac{1}{2(1+N)} + ...
-2
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0answers
24 views

ODE's o infinite order

In the article "LINEAR DIFFERENTIAL EQUATIONS OF INFINITE ORDER" by Carmichael, there is a theorem giving the general solution to infinite differential equations with constant coefficients of the ...
0
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1answer
36 views

When is right to kill $r^l$ and/or $r^{(-l-1)}$?

When we solve the Laplace equation in spherical polar coordinate, we get the radial part whose solution is: $$R=Ar^l+Br^{-(l+1)}$$ Now, some solutions keep this two terms, but when we derive the ...
0
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1answer
39 views

Solving a system of ODE that arose in solving Burgers' equation

Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's ...
2
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1answer
32 views

Show that a piecewise function of two solutions to an ODE is a solution to the ODE

Let $x=x(t)$. If the first order ODE $x'=f(t,x) (*)$ is satisfied by $u$ and $v$, each over $I = (a,b)$, show that $$w(t) := u1_{(a,t_0)} + v1_{[t_0,b)}$$ satisfies $(*)$, where $u(t_0) = v(t_0)$ ...
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0answers
34 views

how to derive exponential growth equation from stochastic growth?

consider an exponential growth process of a population starting that has initial size $N_0$ and grows at rate $r$: $$\frac{dN}{dt} = rN$$ assuming deterministic and constant growth, the population ...
3
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1answer
70 views

Asymptotic relation for the following series?

Questions Is the asymptotic relationship correct? How do I determine $c_1$ and $\kappa$? As, $|s| \to 0$ $$ \sum_{r=1}^\infty s^r \ln(r) \sim c_1 \sqrt{s} + (\kappa - 1 + \frac{\ln(2 \pi)}{2} ...
1
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2answers
30 views

Do we have uniqueness of solutions to the IVP $\dot{x}(t) = x(t)^2, x(0) = 0?$

Consider the following nonlinear initial value problem: $$\dot{x}(t) = x(t)^2, \qquad x(0) = 0.$$ Clearly, one solution to the above equation, which holds for all time $t \in (-\infty, \infty)$, is ...
0
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1answer
33 views

Derivative of determinant. [duplicate]

I have the following identity which I want to prove: $$\frac{d}{dt} det(A+tB)|_{t=0} = Tr( Cof(A)^TB)$$ where $Cof(A)$ is the cofactor matrix of $A$, and $A$ is an $n\times n$ complex matrix. The ...
0
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2answers
41 views

Linear differential system of Bessel equations

I have the following system: \begin{cases} g_{1}^{\prime\prime}+\rho^{-1}g_{1}^{\prime}-(1+\rho^{-2})g_{1}-f_{3} & =0\\ f_{3}^{\prime\prime}+\rho^{-1}f_{3}^{\prime}-(1+9\rho^{-2})f_{3}-g_{1} & ...
0
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1answer
13 views

How to use the finite difference method to solve a transport equation with a source term?

I am going to use the finite difference method to solve a transport equation with a source term. In order to solve $u_t+u_x=0$, we can use $u_j^{n+1}=j_j^n-\lambda*(u_{j+1}^n-u_{j}^n)$, where ...
3
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1answer
53 views

How could I calculate $\lim_{t\rightarrow 0}\frac{x(t)}{t^{\sqrt{3}}}$ for the following

If $x(t)$ satisfy $t^2x''+tx'+(t^2-3)x=0$ then what is the limit $$ \lim_{t\rightarrow 0}\frac{x(t)}{t^{\sqrt{3}}}$$ It is very important to me, hint or full help, please. I know that the solution to ...
0
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1answer
23 views

A function g defined for all real $x>0$ satisfies $g(1)=1, g'(x^2)=x^3$ for all $x>0$ then find g(4)

Problem : A function g defined for all real $x>0$ satisfies $g(1)=1, g'(x^2)=x^3$ for all $x>0$ then find g(4) Please suggest how to tackle this getting no clue how to proceed tried to form ...
4
votes
2answers
77 views

Can Laplace solve every lineair differential equation?

I'm learning about laplace tranform method to solve lineair differential equations but i'm wondering if laplace transformations can be used to solve every linear differential equations there is. Or ...
0
votes
1answer
28 views

Radius of convergence of the solutions of the differential equation

Justifies that the solutions are analytic functions in $t_0=0$ . Is it possible to determine the radius of cnvergencia series corresponding powers without calculate? $$ (1-t^2)x''-2tx'+a(a+1)x=0$$ ...
2
votes
1answer
30 views

Lyapunov functions for delay differential equations with discontinuities?

I want to show that a set of differential equations with a fixed delay and with finite discontinuities converges to some equilibrium. I've simulated it and used a rather ugly $\epsilon,\delta$ kind of ...
2
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0answers
92 views

External ballistics: Prove that the range is a concave function of the elevation

Consider a projectile moving in a plane. One of many different models for this problem is the following ordinary differential equation \begin{align} x''(t) &= -Ex'(t), \\ y''(t) &= -Ey'(t)- ...
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0answers
15 views

solving byp system in matlab

I have problem to solve this in MATLAB. The equation already in first order system, yet I still confused to write the code and solve it. Or this equation cant be solved by using bvp4c perhaps? ...
1
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1answer
31 views

Numerical Approximation of a Differential Equation

I have the differential equation that models the velocity of a falling object: $$ \frac{dv}{dt}= \frac{c}{m}v^2 - g $$ Where: c= drag coefficient = constant m = mass g = acceleration due to ...
0
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1answer
22 views

Solution to first order linear ODE and “variation of parameters”?

I am reading in my textbook about the derivation of the general solution for $$\dfrac{dy}{dx} + P(x)y=f(x)$$ First, the textbook shows that a general solution for this DE is composed by the sum of ...
1
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1answer
83 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
0
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1answer
55 views

Use Existence Theorem to determine if $x = \frac{1}{4-t^2}$ yields a solution/s to $\frac{dx}{dt} = 2tx^2$. If so, give them. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
0
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0answers
28 views

Why do these IVPs have a unique maximal solution? What is the largest possible domain for each ODE?

Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq \mathbb R^2$ s.t. $D$ is open and connected, is a differentiable ...
0
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2answers
20 views

Leibniz Integral Rule, proving independence of time

I am currently attempting this question: Prove that $Q(t)$ is independent of time. $Q = \int_{-\infty}^{\infty} \frac{1}{2}u^{2} dx $, with $\frac{\partial{u}} {\partial{t}} + u\frac{\partial{u}} ...
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0answers
69 views

Use Existence Theorem to determine if $x - \ln(x) = t^2 + 1$ yields a solution/s to $\frac{dx}{dt} = \frac{2tx}{x-1}$. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
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0answers
49 views

Use Existence Theorem to determine if $t^2 - \sin(t+x) = 1$ yields a solution/s to $\frac{dx}{dt} = 2t\sec(t+x)-1$. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
1
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5answers
78 views

Why does the sum of two linearly independent solutions of a second order homogeneous ODE give a general solution?

The following is a short extract from the book I am reading: If given a Homogeneous ODE: $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+5\frac{\mathrm{d} ...
1
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2answers
96 views

Use Existence Theorem to determine if $e^{tx} + x = t - 1$ yields a solution/s to $\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t}$.

Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq \mathbb R^2$ s.t. $D$ is open and connected, is a differentiable ...
1
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0answers
23 views

Evolution of a closed set under a differential equation

I consider an ordinary differential equation $$ \dot x = f(x), \quad x(0) = x_0 \in X_0 \subset \mathbb R^n. \quad (*) $$ Let $f$ allow for a unique solution on $X_0$ and on the whole time interval ...
2
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0answers
73 views

How do we know that an integral is unsolvable?

I am currently learning intro differential equations. I am confused how one knows that an ODE will not be solvable. It seems that for the most part, the equations becomes "unsolvable" about halfway ...
1
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0answers
55 views

Solving the differential equation $y'=(y^2-1)e^{ty}$

In a practice exam, we're asked to solve the differential equation $y'=(y^2-1)e^{ty}$, with $y(1)=0$. However, the only techniques we've covered are separable differential equations, first order ...
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0answers
40 views

Write the Following in matrix notation

Write the following in matrix notation: $$\matrix{i' &=& \cos(wt)i - \sin(wt)j\\ j' &=& \sin(wt)i + \cos(wt)j\\ k' &=& k}$$ Note: $i, j$ and $k$ are all vectors! Show ...
1
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2answers
61 views

“The order of a differential equation is the highest derivative in the equation”. What's wrong with this statement?

I am asked the following "The order of a differential equation is the highest derivative in the equation". What's wrong with this statement? I've checked my text and several other sources, and ...
0
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0answers
31 views

Stability - Which is the characteristic equation?

Consider the system $$\frac{dx}{dt}=-2x+y\sin (xy), \\ \frac{dy}{dt}=-2y+x\sin (xy), \\ \frac{dz}{dt}=-2x$$ I have to show that the equilibrium point is asymptotically stable. $$$$ The ...
0
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0answers
19 views

Stability of the equilibrium point

I have to check the stability of the equilibrium point of the system $$\frac{dx}{dt}=-x+\ln (|t|+1)y-\sin (t)z, \\ \frac{dy}{dt}=-\ln (|t|+1)x-y+e^tz, \\ \frac{dz}{dt}=\sin (t)x-e^ty-z$$ $$$$ The ...
2
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0answers
20 views

Differential equations and Vector spaces

I was reading Cohn's book on Lie Groups.In introduction part he has given the motivation behind Lie Groups.It is like this If solution of the differential equation $\frac{dx_{i}}{dt}=u(t)$ is ...
1
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1answer
21 views

Construction of Lyapunov function

Consider the system $$\left\{\begin{matrix} \frac{dx}{dt}=-2x\\ \frac{dy}{dt}=-2y-2z^2\sin y\cos y\\ \frac{dz}{dt}=-2z(\sin y)^2 \end{matrix}\right.$$ I want to show that the equilibrium point is ...
1
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0answers
28 views

Why is causality important for laplace transformations? [closed]

Could someone please explain why causality is important for laplace transformations?
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0answers
21 views

A problem about Frechet derivative:

Let A be a $n \times n $ matrix with real entries and eigenvalues with strictly negative real parts. Let $g \in C^1(R^n;R^n)$ with $ g(0) = 0 $ and with Frechet derivative $ Dg(0) $ satisfying ...