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

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Proof of a differential equation using chain formula

Let's denote $\frac{D}{Dt}$ as: $$\frac{D}{Dt} = \frac{\partial}{\partial t}+ \sum v^j \frac{\partial}{\partial x_j}$$ Let $h(x,t),v(x,t)$ be a vector fields for which: $$\frac{Dh}{Dt}=h\nabla v$$ ...
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1answer
53 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
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1answer
49 views
+250

How to solve the following system $\frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C$, $ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F$

Is there a way to analytically solve the following ODE system? $$ \frac{\text{d}x}{\text{d} t} = -Ax + B\left(\frac{1}{y} -1\right) \\ \frac{\text{d}y}{\text{d} t} = -Cx + D\left(\frac{1}{y} ...
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0answers
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How to solve Poisson's equation using Green's function in this case?

Let's consider the problem of determining the electrostatic potential given a charge distribution $\rho : U\subset \mathbb{R}^3\to \mathbb{R}$. In that case, the potential satisfies the Poisson ...
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0answers
8 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.
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0answers
13 views

Strong Solutions to Nonlinear ODE by Contraction Mapping

Consider the $1$-d ODE $$-u_{xx}+u-\epsilon u^{2}=f, \tag{1}$$ where $f$ is a nice RHS, say $f\in\mathcal{S}(\mathbb{R})$, and $\epsilon>0$. By using the Bessel potential, one looks for solutions ...
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3answers
36 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 $t$?
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0answers
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Does this fact contradict the Picard-Lindelof uniqueness theorem?

For the equation $$y'' = \frac{(y')^2}{y} - 1$$ there are two solutions $y_1 = 1+ \sin x$ and $y_2 = (\frac{x}{\sqrt{2}} + 1)^2$ passing through the point $(0,1)$. Does the fact contradict the ...
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3answers
379 views

Why does using elimination in a system of first order differential equations produce an incorrect result?

For example, if I have the system, $$ y'+y=3x \\ y'-y=x $$ I could then use elimination to minus the top equation from the bottom one to get, $$ 2y=2x \\ y=x $$ Which is obviously wrong as then, ...
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2answers
47 views

Solve $2x^{3}ydy+(1-y^{2})((xy)^{2}+y^{2}-1)dx=0$

The question is to solve this $$2x^{3}ydy+(1-y^{2})(x^2y^{2}+y^{2}-1)dx=0$$ What I tried was to bring this equation in linear differential equation form but failed. I have also tried out rewriting ...
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1answer
43 views

Solution of $(x^2+f^2(x))f'(x)=1,\forall x\in [1,+\infty)$

A question of interest that arose during my doing a seemingly easy exercise is: The exercise considered a function $f:[1,+\infty)\to\mathbb{R}$ differentiable with $f(1)=1$ and $f'(x)\left( ...
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2answers
29 views

Logistic differential equation problem

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I'm having trouble turning the ...
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1answer
29 views

On a Cauchy problem exercise.

I can't seem to find the trick to solve the following Cauchy problem: \begin{cases} y' = \alpha( 1 - y/ \beta) y \\ y(0) = y_o \end{cases} where $\alpha$ and $\beta$ are greater than zero. Anyone ...
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1answer
18 views

ODE's with Trigonometry functions

I have been given the differential equation $$\frac{dy}{dx}(\tan x) + 2y = x(\operatorname{cosec} x)$$ When $y = 0$ when $x = \pi/2$ Can anyone help me out in this question as I have not done many ...
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1answer
30 views

How long for tank to drain completely?

I tried integrating it: $\frac{2}{5}H^{5/2} = -ct + k$, where $k$ is the constant from integration $H = \sqrt[5]{\frac{25}{4}(k-ct)^2}$ When $t = 0$, $H = \sqrt[5]{\frac{25}{4}k^2}$ When $t = ...
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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 ...
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1answer
20 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
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1answer
28 views

How do I solve the following differential equation (It's not seperable)?

I'm trying to Solve the following equation: find the solution $y:(-1,1) \rightarrow \mathbb{R}$ of $y'=\dfrac{y}{1-x^2}+x$? It is not separable and I have no other Tools to solve it.
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1answer
31 views

How to solve this differential equation which has a square of the 1st derivative?

$Ay'' + B(y')^2 + C = 0$ where $A, B, C$ are constants. I checked that 2nd order equations of the form $Ay'' + By' + Cy$ can be treated as $Ay^2 + By + C = 0$ and then proceeded further. But how ...
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1answer
17 views

Solving Ordinary Differential Equation involving trig functions

I have been given the differential equation: Dy/dx(tanx) + 2y = x(cosecx) Where y = 0 when x = π/2 I don't know where to start with this, and haven't done many ...
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0answers
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Help me find the solution to the IVP in implicit form?

I am having a problem with the (2).Please help me. Text-book Question: Consider the IVP: $$\color{crimson}{(105\sin(3y-15x)-y)dx+(-21\sin(3y-15)-x+2y)dy=0}$$ ...
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1answer
22 views

Proving that $y(t)\to0$ given a dynamical system

Consider a nonlinear system of the form $$\dot{y}(t)=p(y(t)) + u(t)$$ where $$p(q) = a_kq^k+a_{k-1}q^{k-1}+\ldots+a_1q$$ $$u(t) = -\left(\alpha_ky(t)^k+\ldots+\alpha_1y(t)\right)-y(t)$$ with ...
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1answer
10 views

Proving global exponential stability of a perturbed system

Consider the system $$\dot{x} = \left(A + \frac{1}{2\varepsilon}BB^TP\right)x + Dg(t,y),\quad y=Cx,$$ where $g(t,y)$ is continuously differentiable and satisfies $$\Vert g(t,y)\Vert_2 \le k\Vert ...
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0answers
12 views

Deduce stochastic differential equation

Let $X$ be a stochastic process with $dX_t = \alpha X_t dt + \sigma X_t dW_t$ and $Y$ a stochastic process with $dY_t = \gamma Y_t dt + \delta Y_t dV_t$, where $W$ and $V$ are independent ...
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0answers
15 views

Cubic First order ODE

How does one solve the following equation: $$v'(t) = \lambda v(t)(v(t)-a)(1-v(t))$$ for $v$ with $\lambda$, $a > 0$ ?
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7answers
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What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
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2answers
35 views

Initial value problem $y'=f(t)y(t).$

Consider the Initial value problem $$y'=f(t)y(t),y(0)=1$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function. Then the IVP has $1$ Infinitely many solutions for some $f.$ $2.$ A ...
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2answers
33 views

Differential equation $y'=(1+f^{2}(x))y(x)$

Consider the Differential equation $$y'=(1+f^{2}(x))y(x), y(0)=1,x\geq0$$ where $f$ is a bounded continuous function on $[0,\infty).$ Prove that the given ODE has unique solution $y$ and ...
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0answers
21 views

Logistic model - solution verification

I'm looking at the Logistic model: $$\begin{cases} \dot{X} = X(1-X)\\ X(0) = X_0 \end{cases}$$ where the phase space is $M = \mathbb{R}$. The solution appears to be $X(t) = \dfrac{1}{1 + ...
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1answer
14 views

Why does the square of the norm of the fourier bessel series $\|J_n(\alpha_ix) \|^2=\frac{b^2}{2}J^2_{n+1}(\alpha_ib)$ when $J_n(\alpha b)=0$.

The square norm in the coefficient of the fourier bessel series (where x=the weight function) was solved in a proof using: $$2\alpha^2\int_0^{b}xJ_n^2(\alpha x)dx=\alpha^2b^2[J_n'(\alpha ...
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0answers
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Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
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1answer
19 views

What does the function $n(\gamma , z_{0})$ denote in this version of Cauchy Integral Formula?

In my lecture notes, the Cauchy Integral Formula for complex integrals is defined as $$ \int_{\gamma} \frac{f(z)}{z - z_{0}} dz = 2 \pi i \cdot n(\gamma , z_{0}) \cdot f(z_{0}) $$ What does the ...
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1answer
21 views

Harmonic Numbers give Odd Constant

When I was examining the product, (1) $\prod _{i=1}^{\infty } H_{i} ^{(i)}$ I noticed that product converges to a single constant, 1.6798, and does so quite quickly. Is this constant something ...
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0answers
29 views

How to solve an initial value problem consisting of a matrix? [on hold]

I am doing differential equations problems and I came across a problem that I am struggling with. The problem states.... Solve the initial value problem \begin{align} X'(t) &= \begin{pmatrix} 1 ...
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1answer
996 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
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0answers
14 views

System of 2nd Order Non-linear ODE

I am trying to solve a system of 2 equations in Matlab. Both equations are 2nd order, nonlinear ODEs. I don't know which function(s) should I use? $$\ddot{X} = {{-M_{cp} r \ddot{\theta} \cos \theta + ...
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0answers
32 views

Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
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1answer
22 views

To Find Orthogonal Trajectories Equation.

The question is that "A family of curves consists of ellipses with a common axis of length 2. Find an equation of the orthogonal trajectories for this family." There is no given equation for a ...
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0answers
30 views

Green's function and solutions corresponding to a delta sequence

I've been studying Green's functions and there's one situation I've found I don't know really well how to tackle. It is the following exercise: Let $u_{H}(x,t-\tau)$ be the solution of the ...
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1answer
29 views

Green's function for fourth order ODE

I'm studying Green's functions and up to now I've learned how to find it for second order ODE's. That is, given one ODE $$Ly = f(x),$$ with some boundary conditions, to construct $G$ we solve ...
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0answers
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Poincare' map to find periodic solution.

Consider the equation $\dot{y} = (acos(t) + b)y - y^3$ $a > 0, b>0$. I know that I need to recast the equation as a first order system $\dot{y} = (acos(x) +b)y - y^3, \dot{x} = 1$. Also, we are ...
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0answers
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Solving Second order Differential equation.

To solve Second order Differential equation. $$ y''-2y'+4y' = 0$$ $$ y(0)=0$$ $$y'(0)=1 $$ by the quadratic formula, I just found $ r = 1+\sqrt3 i$ and $$ C_1 = 0, C_2=\frac{-1}{\sqrt3}$$ So my ...
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1answer
18 views

Solving an ODE using Picard's Iteration Method

Find the exact solution of the IVP $y'=y^2$, $y(0)=1$ Starting with $y_0(x)=1$, apply Picard's method to calculate $y_1(x),y_2(x),y_3(x)$, and compare these results with the exact solution. ...
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1answer
41 views

Applications of the wave equation

I've recently started to take interest in PDEs and how to solve them, and I'm wondering a bit about real life applications of the wave equation. So far I haven't found anything about practical ...
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0answers
39 views

Solve the dynamical system in polar coordinates

I have the system (it is time dependent, this is a simplified notation): \begin{cases} x' = x - y - x^3 \\ y' = x + y - y^3 \\ \end{cases} I can't seem to solve it for r, $\theta$. (The change of ...
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2answers
35 views

Change of variable to solve boundary value problem

We are given the differential equation $$u''(x)-\lambda u(x)=0$$ (negative $\lambda$), with boundary conditions $u(0)=u(1)=0$. As the solution will involve trig functions I'd like to do a change of ...
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1answer
20 views

Solving a separable differential equation: $xy' = (1-4x^2)\tan(y)$

How to solve $\tan(y)$ to $y$? I can't solve this. $$ xy' = (1-4x^2)\tan(y)$$ $$ x \frac{dy}{dx} = (1-4x^2)\tan(y) $$ $$ \int \frac{1}{\tan(y)} dy = \int \frac{(1-4x^2)}{x} dx $$ $$ \ln|\tan(y)| = ...
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0answers
14 views

Series Solution in ODE?

I had a question about how to find a series solution regarding the following differential equation. $$x(x-1)y''+6x^{2}y'+3y=0$$ I have already found that the roots of the indicial equation are ...
2
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4answers
422 views

Particular solution of $\sin^2(x)$

I have the differential equation: $$y'' + y = \sin^2(x)$$ and to solve it I need to use variation of parameters and therefore I need to find the form of the particular solution. What is your way of ...
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0answers
28 views

Alpha representation of probability distribution

Probability vector is an $n$-dimensional vector $p=(p_1,...,\ p_n)$ that the sum of whose components equals one, i.e. $p_1+...+p_n=1$. If we take the square root of each component of probability, we ...