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

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(Integration) Physics problem to find the time before a collar comes to rest on a ring with friction.

I need to find $\tau$ from this integration/differential equation but I'm stuck.
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Extremizing the boundary value problem $I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$

Extremizing the boundary value problem $$I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$$ My Thought: First we use Euler-Lagrange equation and solving we get , $y(x)=C_1x+C_2$. Then we put it in $I[y]$ ...
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0answers
33 views

Good book for an introduction to differential equations

I've looked at differential equations, but nothing past what you would learn in Calculus I and II (slope fields, separation of variables, etc.). Is there a good book out there which provides a ...
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1answer
21 views

Change of variables for heat equation

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of ...
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3answers
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Does this technique for solving an ODE generalize?

Apologies for what's probably a silly question to anyone who knows this stuff. I was looking at a question earlier and realized that $\sin(x)$ and $\cos(x)$ satisfy two different differential ...
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3answers
44 views

Definite integral using $u$-substitution.

Evaluate the definite integral: $$\int_{1/3}^{\sqrt{2}/3} \dfrac{1}{x\sqrt{9x^2-1}}{dx}$$ So, I am to use $u$-substitution, and immediately, it would appear that perhaps the integral may be some ...
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Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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17 views

Are there functions that are not of exponential order for which you can define a Laplace transform?

I'am in a course of Introduction to Linear Differential Equations and teacher made us this question in class. we work in $\mathbb{R}$, and any help to answer this is welcome
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1answer
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extending real solution to holomorphic function

Given a rational functions $f$ and $g$ defined on an interval $]0,a[$, $a>0$, a function $h$ satisfies in $]0,a[$ the linear differential equation: ...
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Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
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1answer
47 views

How do I prove a differential operator has no purely imaginary eigenvalues?

Anyone who has taken a course in linear algebra knows how to prove the eigenvalues of a self-adjoint operator are real or the eigenvalues of a skew-self-adjoint operator are purely imaginary. This is ...
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WKB for a sixth order eigenvalue problem

I have the following 6th order eigenvalue problem: $$ (D^2 - \alpha^2)^3 y(x) = -\alpha^2 \lambda Q(x) \, y(x), \quad 0 < x < 1, \quad \text{+ BCs}, $$ where $D = \mathrm{d}/\mathrm{d} x $, ...
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1answer
29 views

Algebraic Curves and Second Order Differential Equations

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve. I am aware that the Weierstrass $\wp$ - ...
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2answers
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Solving characteristic equations…

Can someone explain to me what my teacher is doing? $x^2 - ax - b = 0$ ..? Isn;t he using the quadratic formula to solve this problem? If that's the case, then where is the $c$ at? Shouldn't he have ...
2
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2answers
19 views

Number of Real roots of cubic

I just have a quick question about a polynomial and its roots, For example, I was solving the differential equation $$\frac{dy}{dx}=\frac{3x^2+4x+2}{2y-2}$$ I solved it using the basic methods of ...
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1answer
22 views

Derivative of the solution of a IVP

For $f \in C^1(D, \mathbb{R}^n)$, $D \subset \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^{n_p}$ compact, there exists unique solutions (locally) for $\dot{y} = f(t,y)$, $y(t_0) = y_0$. We denote ...
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1answer
39 views

IVP to $y'=\frac{2xy^2+2x}{x^2+1}$

Find a solution of $y'=\frac{2xy^2+2x}{x^2+1}$ with given IVP $y(0)=\sqrt{3}$. My solution: $\int \frac{1}{y^2+1}dy=\int \frac{2x}{x^2+1}dx$ $\Rightarrow \tan^{-1}(y)=\log(x^2+1)+c, c\in ...
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The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
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How to find $ Y(t)$? [on hold]

Consider the IVP in $\mathbb R^2$ $Y^{'}(t)=AY+BY$; $Y(0)=Y_0$ where $A=$ $$ \begin{matrix} 1 & 0 \\ -1 & 1 \\ \\ \end{matrix} $$ and $B=$$$ ...
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0answers
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Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
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2answers
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Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
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Particular solution of a second order differential equation

I have a general (maybe naive) question. Let's suppose I have the following diff. eq.$$y''(x)+a~y'(x)=b(x)$$ To solve it, I cand do a change of variable and write: $$u'(x)+a~u(x)=b(x)$$ Then, if I ...
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ordinary differential equations

I am trying to understand how the solution of this equation goes: $$\frac{y^2-1}{y}\cdot \sin(x^3)=\frac{dy}{dx}$$ with initial condition $y(0)=-0.5$ I would like to understand if the solution can ...
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1answer
25 views

Undamped motion

I want to find out $x$ and $x''$$=\frac {-k}{m}x$ where $\frac km = w^2$. I have parameters $$ \begin{cases} \ x = 0 \\[2ex] x' = V_0, & \end{cases}$$ I can understand that $$x' = \sqrt{V_0^2 - ...
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1answer
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How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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Book on Ordinary Differential Eqnns

What is a complete book containing all the topics listed below : 1.Ordinary Differential Equations (ODEs): Existence and uniqueness of solutions of initial value problems for first order ordinary ...
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1answer
43 views

Solving a differential equation [on hold]

Could someone please explain to me how the following differential coefficient comes about? \begin{equation*} \frac{d(ak^2)}{dk} = 2ak\end{equation*} Thanks :)
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1answer
13 views

Initial value problem-Picard's conditions

This is a true or false test: For b) I don't understand how to find if a differential equation satisfies Picard's conditions or not. Any help would be much appreciated.
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global manifolds

Can you also explain why the global stable manifold is the union of the flow of the local stable manifolds for t < 0? Why do we not include all t?
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1answer
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Differentiation of multiple variables

Could someone please explain how the solution was obtained to the following differential expression? \begin{equation*} \frac{d(VK)}{dK} = V + K\frac{d(V)}{dK}. \end{equation*}
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$f(t,x)$ defined and continuous implies $dx/dt=f$ has solution

Let $f(t,x)$ be defined and continuous for $a\leq t\leq b$ and $x \in \mathbb{R}^n$. Show that the problem \begin{equation} \begin{cases} \frac{dx}{dt} &=f(t,x) ,\\ x(a) & = x_0 ...
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1answer
22 views

PDE Manipulation - Calculus

I need help for this question, its a lot of calculus but I'm confuse. let $$ u= \dfrac{(x-b)^{2}+y^{2}-q^{2}}{(x-b-1)^{2}+y^{2}-q^{2}-1} $$ I need show that $$ u_{x}^{2}+u_y^{2}= ...
3
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1answer
17 views

Initial Value problem and showing that $ x(t_n) - x_n = 1/2ht_ne^{tn} + O(h^2)$

I have a question which I can't for the life of me figure out. The questions starts by giving $$x'(t) = x(t), x(0) = 1, x(t) = e^t$$ So they give you the solution and basically ask you to apply the ...
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1answer
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What does a 3D periodic solution of a differential equation look like?

The Pointcare-Bendixson Theorem implies that if a solution stays in a bounded region with no equilibrium points then it is either a periodic solution or it approaches a periodic orbit as t goes to ...
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Expressing the Solution to a System of Differential Equations

My professor wrote the solution to a system as $$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$ Where the column vectors are the ...
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1answer
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seperable differential equation question

b) $(2xy^3)dx + (3x2y^2 + y^4)dy = 0$ c) $(2xy^3)dx + (3x2y^2 + y^2)dy = 0$ I know that $c$ is a separable differential equation but $b$ is not. Why? The only difference is the power of the ...
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33 views

How to solve this ODE numerically?

I have a question about how to solve this ODE numerically: $$\frac{C}{4}y'^2+\frac{C}{4}y''y+(0.098)^2y''y'''=0$$ where $C$ is a constant and the initial conditions are $y(0)=y''(0)=0$ and $y'(0)=1$. ...
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0answers
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Clarification of Fuchs's theorem

Here is Fuchs's theorem My professor has been saying the last couple of classes that if $p(t)$ and $q(t)$ are polynomials, then the second order differential equation converges everywhere. He hasn't ...
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1answer
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ordinary differential equations of second order [on hold]

How one can solve ODE in the following general form? $$f(x)\frac{d^{2}y}{dx^{2}}+g(x)\frac{dy}{dx}+h(x)(\frac{dy}{dx})^{2}=0$$ where $f$, $g$ and $h$ are continous functions of $x$. Thanks.
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1answer
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Non-Dimensionaliztion of coupled equation

I'm going over some review for a project I'm doing over the summer and ran into a problem of non-Nondimensionalization. I have not done it in a while and am struggling on how to approach this problem. ...
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1answer
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Bounds of ordinary differential equation

I need to show that the solution for the following equation $$\ddot{x}=-\log x-1$$ is bounded for every initial condition. I started by converting to the system $\cases {\dot{x} = y \\ \dot{y}=-\log x ...
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Question about proof of unique solution of differential equations

Theorem: Let $a' > 0$, $b > 0$, $(x_0, y_0) \in \Bbb R^2$, $\Bbb R' = \{(x, y) : |x - x_0| < a', |y - y_0| < b\}$, $g: \Bbb R' \to \Bbb R$ continuous on $\Bbb R'$, and for some $k > ...
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1answer
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Is this a property of commutative differential operators?

I found this claim in a paper but the proof escapes me. I'm sure that it's simple. Suppose we have $\psi$, a solution to the ODE $LQ\psi=0$, where $L$ and $Q$ are commutative differential operators. ...
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3answers
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Solving seemingly easy differential EQ

I have the differential equation $2\frac{d^2\phi}{d^2\zeta}=e^{-\phi}$, where $\phi=\Phi/\sigma_z^2$, $\zeta=z/z_0$. I also have the fact that $ln(\rho/\rho_0)=-\Phi/\sigma_z^2$. Given the boundary ...
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ODE with with translated arguments

I'm working on a problem of Approximation Theory and to move foward I have to solve the following ODEs: \begin{equation} -2\pi iw.e^{-2\pi itw} = y'(t) + y(t-\frac{1}{2}) \\ y(-1/2) = 0, \\ -2\pi ...
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1answer
29 views

Differential Equation (Non linear to linear differential equation)

Show that the substitution $u=\frac{1}{y}$ transform the non-linear differential equation $$\frac{dy}{dx}+\frac{y}{x}=y^2\ln (x)$$ into the linear differential equation ...
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1answer
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How to prove this ODE is stable but not asymptotically stable?

Consider the ODE in polar coordinates: $$ r'=f(r),\theta ' =1 $$ where $$ f(r)=r\sin (1/r^2), r\neq 0, f(0)=0. $$ show that the origin is stable but not asymptotically stable.
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1answer
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How to solve a non-linear ODE

I need to find the solution of the equation: $$(2e^y-t) \dot{y} = 1$$ with the initial value $y(0)=0$ Here's what I tried: I tried to separate y and t - didn't work. I tried to denote $x=2e^y$ and ...
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Nondimensionalization of complex nonlinear ODE

I am interested in obtaining the nondimensional form of the rather complicated complex first order ODE $$\left(i-\frac{1}{2\Omega}f_{m,n}\right) \frac{d a_{m,n}(t)}{dt} =E_{m,n}^{\text{kin}}(t) + ...
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Deriving equation for sequential decay?

The differential equation describing the decay of a particle (p1) into another particle (p2), which then decays into a further particle (p3) is: where is the number of p2 particles, and is the ...