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

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32 views

Heat equation-unicity

We have the folllwing problem: $\begin{cases} & \dfrac{\partial u}{\partial t} = k \dfrac{\partial^2 u}{\partial x^2}, 0 < x < l, t > 0\\ & u(0,t)=0,\\ & \dfrac{\partial ...
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2answers
28 views

Change of variables in multivariable differential equations

This is a very easy question about how to justify the change of variables. Let $f$ be a $C^1$ function of two variables $x,y$. Introduce the variables $s,t$ as: $$\begin{cases} s=x+y \\ t=x-y ...
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3answers
40 views

Gaining some insight about Picard–Lindelöf theorem.

In class I have been introduced to the Picard–Lindelöf theorem. It was written down in all its technical glory.Now: the important things to remember from it were that if a function is continuous; and, ...
0
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1answer
25 views

Can't understand a notation regarding weak solution of Vlasov-Poisson system

The text is from https://cmouhot.files.wordpress.com/2010/01/chapter5.pdf . In section 1, it uses $~f_t~$ to represent a smooth solution of Vlasov-Poisson system (VPS). I think here $~t~$ in $~f_t~$ ...
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3answers
37 views

solving second order non-homogeneous differential equation 2

I have a problem on solving this differential equation. $ y=c_{1}+c_{2}e^{-2x}-cos2x-1/2(sin2x-cos2x) $ I reached : $ y''+2y'=4sin2x $ as answer but I'm not sure. my step by step solution is : $ ...
4
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1answer
73 views

Backwards Heat Equation $ u_{t} = -\lambda^2 u_{xx}$

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} & = \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
3
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1answer
49 views

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
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0answers
21 views

is 1/x a lipschitz continuous function? [on hold]

Actually, I would like to know if the dynamics associated, with initial condition $x(0)=1$, has any solution in $\mathbb{R}^2$. I tried to find the solution and I got that $x=\pm\sqrt{2t+1}$, but then ...
0
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2answers
38 views

A question about $ y' = \frac{-x^3y}{(x^2+y^2+1)^2} $.

Given the differential equation: $ y' = \frac{-x^3y}{(x^2+y^2+1)^2} $, I need to prove that if $y(x)$ is a solution for the equation, then $y$ is an even function. thus, what can we say about the ...
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0answers
20 views

Differential Partial equation with fourier transform [on hold]

how to apply the Fourier transform to the following problem $P:y''+4y=0$ with boundary conditions $y(0)=0=y(\pi)$
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3answers
46 views

How to find differential equation

$$\frac{dy}{dx}-8x=2xy^2\quad y=0\,x=1$$ I separated $x$ and $y$. \begin{align*} \color{red}{\frac{dy}{y^2}}&=\color{red}{2x+8x dx}\\ \frac{dy}{y^2}&=\color{red}{10x dx}\\ \color{red}{\ln ...
0
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1answer
24 views

Solve differential equation [on hold]

Help me solve this equation $$u_{tt} + 81u_{xx} = 0 (x \in \mathbb R, t\ge 0)$$ $$ u(x,0) = x^3 \sin(x)*\chi_{[-\pi, \pi]}(x) , u_t(x,0) = 0$$
2
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2answers
72 views

Solution of differential lyapunov equation

How would I solve for following, else any implemented algorithms or solvers in matlab (even ways to solve it) for Lyapunov differential equation of form: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
2
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1answer
368 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
0
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1answer
26 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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0answers
35 views

PDE complex boundary condition

My attempt to this question was setting T''-lambda T=0 and try lambda=0, >0 and <0. However, I do not seem to have sufficient information to determent which case have non-trivial solution ( since ...
1
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3answers
53 views

I've got the following differential equation, how do I integrate the expression to get the answer?

-Solve the differential equation ,with the given condition: $${\partial z \over \partial x}+(2e^x-y){\partial z \over \partial y}=0.\ \ z=y\ \ \ at \ \ \ \ \ x=0. $$ I solve it as follows: $${dx ...
3
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2answers
56 views

Advice on second order non-linear ordinary differential equation

I'm currently working on some problems concerning the calculus of variations and I have come up with the following differential equation that I now want to solve: $$1 + y'(x)^2 - y''(x)(y(x)-\lambda) ...
0
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1answer
13 views

*scVector Differential Equation

What's the general method for solving differential equations of this type $$\frac d{dt} \pmatrix {y \\ x} = \pmatrix{y_0 \\ x_0} + y\pmatrix{a \\ b}$$ where $y=y(t), x=x(t)$, and $y_0$, $x_0$, $a$, ...
2
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3answers
49 views

How would this differential equation be solved?

How would this differential equation be solved? $$y{\partial z\over \partial x}+z{\partial z\over \partial y}={y \over x}$$ I was taught to solve them like : $${dx \over y}={dy \over z}={dz ...
4
votes
2answers
40 views

Check my general solution to the differential equation?

Given differential equation: $$y' = \frac{(2xy^{3}+4x)}{(x^{2}y^{2}+y^{2})}$$ This is the general solution that I got for the above differential equation: $$\frac{1}{3} \ln{\lvert ...
9
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3answers
212 views

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an ...
2
votes
2answers
145 views

If a solution of an IVP is non-unique, then there are infinitely many solutions.

I am trying to prove than when an IVP has a non-unique solution then there exists infinitely many different solutions. I know that when the lipschitz conditions holds that there is at most one ...
0
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0answers
40 views

Solving differential equation: $\dfrac{dx}{dy}=\dfrac{G(y)}{\sqrt{1-G^2(y)}}$

I'm carrying out an experiment in which I have to solve the differential equations: $$\dfrac{dx}{dy}=\dfrac{A\left(ly-\frac{y^2}{2}\right)}{\sqrt{1-A^2\left(ly-\frac{y^2}{2}\right)^2}}.$$ I don't ...
0
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1answer
28 views

How could we continue to show the inequality?

Let $\Omega$ a bounded space. Let $u_1$ the solution of the problem $$-\Delta u_1(x)=f(x), x \in \Omega \\ u_1(x)=g_1(x), x \in \partial{\Omega}$$ and $u_2$ is the solution of the problem $$-\Delta ...
0
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0answers
15 views

How do I solve this system of PDEs numerically?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} (\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}) f(x,y) = F(f,g,h) \\ (\frac{\partial}{\partial x} - ...
0
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2answers
307 views

Differential equations: Of non-vertical lines

Find the differential equation of all non-vertical lines in a plane. I am not sure how to solve this problem. To do that I have to create an equation. I am not able do so. Is it $x \ne 0 $ ? ...
0
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1answer
24 views

First-order differential equation - integrate between functions

I have to solve this diff. equation: $y'(t)+\frac{y}{\tau}=\frac{x(t)}{L}$ This is a first-order linear differential equation, so, by solving it, I get: $y(t)=e^{-t/\tau} \int e^{t/\tau} ...
1
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1answer
18 views

Particular Solution of second order Linear Differential equation

I've got a problem with finding particular solution of: $$y''-y'-6y=12x.$$ My homogeneous solution is : $$y=C_1e^{3x}+C_2e^{-2x}.$$ When i'm trying to find particular solution i'm using the Method ...
0
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2answers
18 views

Derivative of: $x\log_2(x)$

Can someone please help me with the derivative of this function: $$y = x\log_2(x)$$ This is the answer: $$1+\ln(x)\over\ln2$$ When I try to solve it I get stuck here: $$\log_2(x) + {x\over x\ln2} ...
0
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1answer
34 views

Determining the maximum intervall of existence

Can somebody explain to me the concept of an intervall of existance/interval of validity? Is it basically the domain of my differential equation? I tried to look it up and I came across this site ...
0
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2answers
19 views

Method of Undetermined Coefficients - Efficient Way of Finding Constants

$$y'' - 4y = (x^2 - 3)\sin(2x)$$ I have to solve the following differential equation. I have gotten everything setup. I have two questions however. Is the form of the particular solution: $y_p = ...
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3answers
22 views

Another confusion about initial condition in ODE

I am still having some confusion on certain problems. Here is what I mean, the questions asks to solve the IVP and determine how the interval on which the solution exists depend ends on the initial ...
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1answer
278 views

acoustics under water

I've got the following problem that is taken from the numerical analysis book by Kahaner-Moler-Nash (P8-15): The speed of sound in ocean water depends on pressure, temperature and salinity, all ...
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0answers
13 views

Prove existence of unique maximal solution

I need to prove that $x'=t^2\sqrt{1+2x},~ x(0)=0$ has a unique maximal solution. I thought of using that $\sqrt{1+2x}$ is bounded by a linear function, but the only theorem I know about that needs ...
2
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0answers
28 views

Prove the $f_1, f_2$ is a basis of linear subspace of solution of differential equation

Let $p,q \in C(\mathbb{R}), L_{pq} = \{f \in C^2(\mathbb{R}):f^{(2)} + pf^{(1)} +q f = 0\} $ For each $(a,b)^T \in \mathbb{R}^2$ there is only one $f \in L_{pq}$ with $(f(0),f'(0)) = (a,b)$ 1- ...
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0answers
21 views

Classification of Differential Equations

I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain. I also know ...
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1answer
36 views

Converting a non-linear ODE to a Bernoulli equation

I am self-studying differential equations using MIT's publicly available materials. The first part of one of the recitation exercises runs as follows: Show that \begin{align} (3e^{2y}x^{\frac{2}{3}} ...
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2answers
59 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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3answers
33 views

Algebraic Manipulation of a Separable Equation

$$(sin(x)+x^2e^y-1)dy/dx=-ycos(x)-2xe^y$$ I understand how to do these problems but I'm wondering how do I get the y variables on the left hand side and the x variables on the right hand side so that ...
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2answers
17 views

Brief question in regard to existence and unique (ODE)

I am just having a bit of trouble understanding the answer to this problem. It asks where in the ty plane would the ODE satisfy the existence and uniqueness theorem, that is; $\mathbf{Thereom}:$ ...
2
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1answer
22 views

$y''(x)+2 x y'(x)+\left(x^2+1\right) y(x)=0$

I constructed this equation so that it would have a double root for $e^{-x^2/2}$. I basically applied $(D+x)(D+x)y$, which gave me this equation. The solution is $c_1 e^{-\frac{x^2}{2}}+c_2 ...
2
votes
1answer
19 views

Find minimizer of the functional

Find minimizer of the functional $ l(u)= \int \limits _{-1} ^1 u(t) \mathbb d t $ with $u(-1)=u(1)=0 $ subject to $g(u)=\int \limits _{-1} ^1 \sqrt{1+u'(t)} \mathbb d t=π $. I solved it using ...
0
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1answer
31 views

Problem on string vibration

Given the standard wave equation for small amplitudes, we have been asked to find the position of string $y(x,t)$, given: $y(x,0)=\sin x$, and, $y'(x,0)=\cos x$, where $y'$ depicts partial ...
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1answer
31 views

Find the integral curve of the differential equation $y'-\frac{2y+1}{x}=1$ belongs point of $M\left(0;-\frac{1}{2}\right)$ [closed]

Find the integral curve of the differential equation $$y'-\frac{2y+1}{x}=1$$ belongs point of $$M\left(0;-\frac{1}{2}\right)$$
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1answer
33 views

How could we continue to get a contradiction?

Let $\Omega$ a bounded space. Using the maximum principle I have to show that the following problem has an unique solution. $$-\Delta u(x)=f(x), x \in \Omega \\ u(x)=g(x), x \in \partial{\Omega}$$ ...
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1answer
37 views

Find the general solution of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$ [closed]

Find the general solution of the differential equation $$\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$$
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2answers
33 views
1
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2answers
32 views

Solving linear differential equation $t\dot{x}(t)+3x(t)=-\frac{1}{t^2+1}$

I want to solve the following linear differential equation (initial value problem): $$t\dot{x}(t)+3x(t)=-\frac{1}{t^2+1}; x(1)=\frac{\pi}{4}$$ I first tried to solve the homogeneous diff. eq. ...
6
votes
3answers
172 views

Approximate solution of differential equation

My task: find approximate solution as $$y = y_0(x) + y_1(x)\lambda + y_2(x)\lambda^2 + y_3(x)\lambda^3$$ of differential equation $$y' = \sin x + \lambda e^y, y(0)=1-\lambda. \ \ \ \ (*)$$ My ...