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

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How to show an ODE system has no global solution

Can the following ODE system has a solution for all real number? $x'(t)=3y^2(t)$, $y'(t)=2cx(t)-1$, here $c$ is a constant. I think it can not be solved for all $\mathbb{R}$. How to show an ODE ...
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0answers
17 views

How much of the chemical will be in the pond after a very long time?

A pond initially containing 1000000 gal of water and an unknown amount of undesirable chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of 300 gal/hr. The ...
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2answers
14 views

Volume estimation with differential equations

The problem reads: "Using differential equations, estimate the volume necessary to build a tube that is 12m long and has an inner diameter of 25cm and an outer diameter of 25,2 cm." Unfortunately I ...
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0answers
15 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...
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1answer
27 views

Prove that $\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$

Prove that $$\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$$ I started out with the following identity: $$ \frac{1}{s}\mathcal{L}(f)=\frac{1}{s}\int_{0}^\infty e^{-st}f(t)dt ...
3
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1answer
32 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
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4answers
58 views

ODE $2yy'' - 3(y')^2 = 4 y^2$

I'm trying to solve the equation by using these substitutions (how it was suggested in my textbook): $$ y = e^{z(x)} \implies y' = z'y \implies y'' = y((z')^2 + z'') $$ The result is: $$ 2y^2((z')^2 ...
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0answers
11 views

Sturm-Liouville eigenvalue problem of order 4

I want to solve the eigenvalue problem $W''''=\lambda W$ with the boundary conditions $W(0)=W'(0)=W(l)=W'(l)=0$. Has someone a hint how to solve that? Thank you...
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1answer
16 views

Solve Sturm-Liouville eigenvalue problem with substitution

I need to solve the SL-eigenvalue problem: $x^4y''+\lambda y = 0$ with $y(1)=y(2)=0$. Therefore one should: 1) substitute with y(x)=xv(x) to get a diferential equation for v(x) and then 2) ...
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2answers
25 views

How would we know that the particle satisfies both cases?

Consider the differential equation $$\ddot{x}=-n^2 x$$ Now it can be shown that an equivalent formula is $$v^2=n^2(A^2-x^2)$$ , where $A$ is the amplitude of this simple harmonic motion and ...
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2answers
36 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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0answers
24 views

How Do I solve the Following equation. Getting Confused.

$ (D^4+2D^2+1)y = x^2 cos x $ I applied Inverse Operator case 5 ie $ q(x)= x^m * cos ax $ = Rational Part of $ e^{iax} $ $ 1\over {f(D+ia)} $ $ x^m $ = Rational part of $ e^{iax} $ $ f(D+ia)^{-1} ...
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3answers
279 views

Non linear Differential Equation

Let $\Omega:=\{(x_1,x_2) \subset \mathbb{R}^2 | x_2>0\}$. I want to solve the differential equation $$\begin{pmatrix} \dot{x_1} \\\dot{x_2} \end{pmatrix}=\begin{pmatrix}x_2^2-x_1^2 ...
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6answers
89 views

Solution of $(x^2 + y^2)\ dx -2xy\ dy$ = 0

Solve $(x^2 + y^2)dx -2xydy = 0$ The answer is $x^2 - y^2 = Cx$ I've tried the following methods but I'm not getting the answer : Variable Separable (n/a) Homogenous Differential Equation ...
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0answers
18 views

The meanings of some symbols in “Calculus of variations”

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked. Thank you very much!!!
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3answers
18 views

Finding a function whose graph passes through two given points, given its (constant) second derivative

It is known that $y(x)$ passes through the points $(0,2)$ and $(1,4)$. Solve for $y(x)$ if the second derivative is: $$\frac{d^2y}{dx^2} = 1 .$$ The answer is: $$y = \frac{1}{2}(x^2 + ...
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0answers
18 views

What is meant by “homogenous problem” exactly?

Let us look at an entirely linear problem with operator $L$. For an algebraic equation $Lu=0$ is a homogenous equation. If $L$ is a differential operator (PDE or ODE) it has to be supplemented with ...
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0answers
20 views

Is this Riccati ODE solvable? If so, how may I guess the particular solution?

I'm working on a problem and came across this Riccati(?) ODE. Is this solvable? Or must I have two other ODEs for $a(t)$ and $\theta (t)$? $m'(t) = - c_1 \frac{m^2 (t)}{a(t)}\cos(\theta (t) ) - c_2 ...
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1answer
17 views

Reversing Implicit Differentiation to determine One Parameter Family of Lines

Determine the orthogonal trajectories of the one parameter family of lines y-Cx = 0; Answer is x^2 + y^2 = C Of course you can always do implicit differentiation on each answer from the set of ...
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2answers
32 views

Continuous compound word problem using ordinary differential equation

I have a problem with one of my homework questions. (b) A certain bank compounds interest continuously at an annualized interest rate $0<r<1$ (measured in inverse-years), meaning that ...
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0answers
17 views

Second order perturbed equation

I've been studying asymptotic behavior on Ordinary Differential Equations. While doing some excercises I found out one excercise which has had me thinking for a while, so I am asking humbly for your ...
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1answer
53 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
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0answers
34 views

A differential equation I

Consider the second order differential equation \begin{align} 2 t^{3} y'' + (5 t^{2} - t) y' + (t^{2} - t + 1) y = 0 \end{align} with the conditions $y(0) = 0$ and $y'(0) = 1$. A solution is known in ...
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1answer
23 views

Use reduction of order to find a solution of the given nonhomogeneous equation.

Question Use reduction of order to find a solution of the given nonhomogeneous equation. The indicated function $y_1(x)$ is a solution of the associated homogeneous equation. Determine a second ...
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1answer
28 views

How to use separation of variables on this differential equation?

Let $a,b,c,d$ be constants. How do I separate $ ay''+b = \frac{c}{(d+y)^3}$ ? I don't need the solution $y=...$, but I need the form $ dy = ... dt$
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0answers
35 views

Second Order Differentials: Using $y = A + Bxe^x$

I've went over some of my math work which I'm currently doing at Uni and came across a rather confusing example. The example I went over is based on Second Order Differentials. So basically what I ...
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1answer
30 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
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1answer
28 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
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1answer
30 views

First ODE problem solution different than WolframAlpha solution

$-y'' +2y' - y = x$ , with conditions $y(0) = y(1) = 0$ I am supposed to find a solution for this problem, so I started with finding the result for the homogeneous equation, and i got $y = c_{1}e^x + ...
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1answer
34 views

Inverting the differential operator $D^2-3D+2$ [on hold]

I am trying to calculate $$(D^2-3D+2)^{-1}(xe^{3x})$$ that is, find a function $f$ such that $(D^2-3D+2)(f)=xe^{3x}$ where $D=\frac{d}{dx}$. Using inverse operator, I am getting an incorrect answer. ...
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1answer
24 views

How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty ...
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1answer
24 views

Conditions of a differential equation

Consider the differential equation \begin{align} 2 x^2 y'' + x(x^2 - 1) y' + (2 x^2 - x +1)y = 0 \hspace{5mm} y(0) = 0, y'(0)=1. \end{align} A solution readily found is \begin{align} y(x) &= B_{0} ...
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0answers
21 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
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5answers
217 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
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0answers
22 views

Phase line and Equilibrium Points

Consider the differential equation $dy/dt=y^8+3y^6-y^2-1$. Sketch the phase line and classify the equilibrium points. Since when $y=0$, the derivative is negative and when $y>1$ the derivative ...
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0answers
7 views

Representing solutions of a second order linear differential equation as the solutions of 2 first order linear differential equations.

Consider $xy''+2y'+xy=0$. Its solutions are $\frac{\cos x}{x},\,\frac{\sin x}{x}$ . Neither of those solutions (as far as I could find) can be the solutions of a first order linear homogeneous ...
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1answer
47 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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0answers
31 views

Trajectories of predator prey equation

I am studying the predator prey equation recently, and here is an example: Let $x'=x(1-0.5y)$ and $y'=y(-0.75+0.25x)$. This is a predator prey equations. Then ...
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0answers
31 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
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1answer
47 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
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1answer
64 views

What is $\int\sinh(x)^pdx$?

What is $$\int\sinh(x)^pdx$$, where $0<p<1$?. I tried using Mathematica, but it came up with some Hypergeometric2F1 function. Is there a simpler answer in this integral?
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0answers
36 views

Can I have variables extreme of integration?

Suppose you have a function $v(t)$ that you want to find. The condition is that it's integral is some fixed quantity. The integral is done between $0$ and $u(t)$, where $u(t)$ is an increasing ...
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0answers
21 views

Stieltjes differential equation

So I have the following differential equation that I want to solve: $$ dy(t) = -d[\alpha(t)\cdot t]\,\,\,\,,y(0) = 50$$ where $[\cdot]$ is the greatest integer function. My guess is that $$y(t) = ...
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2answers
56 views

Finding second derivative for $x=\sin t$ and $y= \sin 2t$.

If $x=\sin t$ and $y= \sin 2t$, how to find second derivative of $y$ w.r.t $x$ ? Or rather how to prove $(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac {dy}{dx}+4y=0$? Is there any shortcuts to find these ...
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0answers
20 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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1answer
18 views

Inverse Laplace Transform and the Unit Step Function

I need to find the inverse Laplace transform of the following function: $$ F(s) = \frac{(s-2)e^{-s}}{s^2-4s+3} $$ I completed the square on the bottom and got the following: $$ F(s) = (e^{-s}) ...
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0answers
30 views

Simple differential equation question [duplicate]

I think this is pretty easy but it's been forever since I've done this. Can someone help me start or give me some tips?
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2answers
50 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla ...
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2answers
14 views

Finding the equation of parabolas with axis parallel to the x-axis

I've seen a post like this but it's on hold and doesn't really help, can someone give me a hint or a step by step solution on how to solve this? I think the other guy is from another section of my ...