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

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0
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3answers
635 views

Finding a solution to dA/dt=-5A if A(0)=5

How does one find a particular solution to this problem? The question states: Find a solution to $ \frac{dA}{dt} = -5A $ if A(0) = 5 Since they give the equivalence of $\frac{dA}{dt}$ A, integrating ...
6
votes
5answers
568 views

Finding a non constant solution to $ (x')^2+x^2=9 $

How do I find a non-constant solution this equation? I've tried to solve for $x$, but the final answer should be in the form of $x(t)=...$ $ (x')^2+x^2=9 $ I'm not sure where to start.
1
vote
2answers
76 views

When can we make a change of variables $f'$ for $f$?

In my applied math class, my instructor introduced the example of two point masses, both with mass $m$, with positions $x_1(t)$ and $x_2(t)$. Newton's law gives us the differential equation $$r'' + ...
6
votes
1answer
118 views

Special functions and diff eq's …

They're are all these methods of dealing with linear second order diff eq's: generating function; recurrence relation; Rodrigues differential form; Schlafi integral form; associated form; second ...
3
votes
0answers
54 views

Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
0
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1answer
107 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...
-1
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1answer
27 views

differential equation order 2 solution-exercice

let $R(t)$ an function on $t$ définied in $\mathbb{R}.$ if we have $2 R'(t)=r(t) R(t)$ and $\dfrac{R''}{R} = \dfrac{r^2}{4} + \dfrac{r'}{2}$ how we found $R?$
1
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1answer
81 views

Show that this relation is an implicit solution of the following differential equation

Differential equation: $$\frac{dy}{dx}=\frac{xy}{x^2+y^2}$$ Relation: $$2y^2 \ln{y} - x^2 = 0$$ From this, I end up getting: $$\frac{dy}{dx} = \frac{x}{2y\ln{y} + y} $$ The missing step would ...
0
votes
1answer
291 views

Wronskian-Differential Equations

The equations below are matrices: Consider the vectors $y^{(1)} (t)$=$\begin{pmatrix}t \\1 \end{pmatrix}$ and $y^{(2)}$ (t)=$\begin{pmatrix}t^2 \\2t \end{pmatrix}$ (a) Compute the Wronskian of ...
1
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1answer
162 views

How to compute Bochner laplacian $\Delta=\nabla^*\nabla=\sum \nabla_{e_i}$?

I'm struggling with proving that Bochner laplacian can be described by the following formula similar to the standard laplacian formula from calculus: $$\Delta = \sum_i \nabla_i^2,$$ where $\nabla_i = ...
1
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2answers
154 views

Tricky differential equation is solveable? $\frac{dy}{dt}=\sqrt{f(t)-y}$

It would be great to solve this problem! But I think maybe not possible because of the square root... is there anything that can be done with this? I guess this is invalid right: ...
0
votes
1answer
147 views

Partial differential equation-delta Dirac and Heaviside function!Solved just need to check please :D

I got 2 questions to ask! I have finished one but not sure if it's correct so I need to double check with someone :) http://imageshack.us/a/img708/1324/83u8.png Here is my worked solution, I took ...
1
vote
1answer
71 views

Maximal Solutions

Let $I\subseteq \boldsymbol{R}$ be a non-degenerated interval and $f:I\to \boldsymbol{R}$ a continuous function and $\forall x\in I$, $f(x)\neq 0$. Given an $a\in I$ let $$\begin{align} ...
1
vote
1answer
100 views

Find a function $f(x) : f(1)=2 \land f'(x)=f(x)^2 \; \forall x$

I cannot find a function that can be derived two times and such that $f(1)=2 \land f'(x)=f(x)^2 \; \forall x$. Could you help?
10
votes
3answers
395 views

Why is it legitimate to solve the differential equation $\frac{dy}{dx}=\frac{y}{x}$ by taking $\int \frac{1}{y}\ dy=\int \frac{1}{x}\ dx$?

Answers to this question Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution? assert that to find a solution to the differential equation $$\dfrac{dy}{dx} = \dfrac{y}{x}$$ we may ...
0
votes
1answer
64 views

Initial Value Problem Question

This is the problem: For the differential equation $y'=$exp$(x−y)$ find the solution for the initial value problem $y(0)=0$. I tried to plug in 0 into the equation, which lead me nowhere - ...
2
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2answers
56 views

Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution?

I have to solve $\dfrac{dy}{dx} = \dfrac{y}{x}$. So I set $v = \dfrac{y}{x}$ and so $$ \dfrac{dy}{dx} = v $$ Then by product rule $x\dfrac{dv}{dx} + v = v$ and so $x\dfrac{dv}{dx} = 0$. But then that ...
0
votes
1answer
47 views

Linear Equation Question

This is the question: If the linear equation $y'+p(x)y=q(x)$ has solutions $y=1$ and $y=x$, what are $p$ and $q$? Am I just supposed to substitute y=1 and y=x into the equation? What I mean is if I ...
2
votes
0answers
123 views

What does 'mod' stand for in this ODE book?

I've seen in a book some: mod(a)=b What does it mean? I've seen this in a book on solving ODE by using symmetries, but I am not sure about what it means. Stephani's Differential Equations: Their ...
1
vote
1answer
73 views

Is $y' + ay^4 + b = 0$ elementary?

I was trying to solve a physics problem getting at the end this differential equation: $$y' + ay^4 + b = 0$$ This isn't a linear differential equation, so I need something tricky. It's really close ...
1
vote
1answer
33 views

Differential equations: solving separable equation

Solve the separable equation $y' = (x-8)e^{-2y}$ satisfying the initial condition $y(8)=\ln(8)$. I can not figure this out I am not sure what I am doing wrong.
2
votes
1answer
132 views

calculus new method

could you help me to find to find a function $f(x)$ so that $$ \frac{d f(x+1)}{d x}+\frac{d f(x+2)}{dx}=\frac{1}{\sqrt{(x+1)^3}}? $$
2
votes
2answers
217 views

General solution to an PDE/ODE, change of variables

OK, this should be a simple homework problem from the text, but I want to be sure I am following the steps through properly because I feel I am missing the very last bit. Given: $$\frac{\partial ...
4
votes
6answers
995 views

New & interesting uses of Differential equations for undergraduates?

I'm teaching an elementary DE's module to some engineering students. Now, every book out there, and every set of online notes, trots out two things: DE's are super-important, vital, can't live ...
0
votes
1answer
95 views

Differential equation system

Solve following differential equations system: $$\overline{x}'(t) = \begin{bmatrix} 3&-2 \\ 4&-1 \end{bmatrix} \overline{x}(t)$$ I don't have answer to this task, so I will be grateful if you ...
1
vote
2answers
335 views

Easy way to find the streamlines

In a textbook, this problem appears: Find the streamlines of the vector field $\mathbf{F}=(x^2+y^2)^{-1}(-y\hat{x}+x\hat{y})$. The system we need to solve, I suppose, is: ...
2
votes
2answers
146 views

Solving $v_{t}+v(x,t)v_{x}=0$ with initial condition

This problem comes from an undergraduate course in PDE. The first question of the problem was to solve the following PDE: $v_{t}+v(x,t)v_{x}=0$ with the following initial condition: $v(x,0)=5x$ ...
1
vote
1answer
69 views

Prove that exactly one solution of ODE converge

$xy'-(2x^2+1)y=x^2$ How can I prove that there is only one solution that is finite when $x\rightarrow \infty$ and how can I find it.
6
votes
2answers
214 views

How to interpret the meaning of “$y$ solves the DE” to have nice properties.

Assume that $I$ is an open interval $0 \in I$ $x$ varies in $I$ $y$ is a differentiable function of $x$. Now in the context of these assumptions, consider the following problem. ...
0
votes
1answer
59 views

Differential equations with power series method

Using power series method solve $$tx''(t) - tx'(t) - x(t) = 0 , \\ x(0)=0, \\ x'(0)=1$$ We can take $$x(t) = \sum_{n=0}^{\infty}a_nt^n$$ Furthermore we have $$x'(t) = \sum_{n=0}^{\infty}na_nt^{n-1} ...
1
vote
0answers
78 views

Convergence of solutions in initial value problem

I am working on the following problem: Suppose $u_{n} : [-M, M] \rightarrow \mathbb{R}$ are differentiable and are such that $u_{n}'(x) = F(u_{n}(x), x)$ for $F$ continuous and bounded. Furthermore, ...
2
votes
3answers
2k views

Finding the values of k for an equation which is a tangent to a curve

I have been faced with the following question: Find the values of $k$ for which $y = kx - 2$ is a tangent to the curve $y = x^2 - 8x + 7$ I managed to figure this out by treating them as ...
0
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2answers
78 views

Question on ODE

Try to find the general solution for $\dfrac{dy}{dx}=3y^{2/3}$ Your answer will probably be $y=(x+C)^3$. Observe $y=0$ is also a solution and it cannot be expressed as $y=(x+C)^3$ for any value of ...
1
vote
1answer
428 views

Eigenvalues and Eigenfunctions of a singular Sturm-Liouville operator using Bessel functions

I’m trying to find the eigenvalues and eigenvectors of the Singular Sturm-Liouville operator: $$Lu=xu''+u'$$ $$u(1)=0$$ $$u(0) \text{ is finite}$$ $$0 < x < 1$$ My approach to solving ...
2
votes
2answers
100 views

Solving for a general solution in $y' = y^2$

My book gives me $y' = y^2$ and then asks me to find the general solution. I am getting $\displaystyle\frac{-1}{(c+x)}$ as my answer. However, both the book's answer key AND Wolfram report to me ...
3
votes
2answers
95 views

Nonlinear ordinary differential equation (Elsgolts)

Please, help me to solve the following non-linear ODEs: \begin{align} \tag 1 y &= (y')^4 -(y')^3 -2 \\ \tag 2 y' &= \dfrac{y}{x+ y^3} \end{align} Thanks.
2
votes
0answers
45 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
2
votes
1answer
34 views

The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$ \frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
5
votes
2answers
142 views

How can we conclude that such function $f(x)$ either exists or not ?

In this PDF file here ( file from OCW.MIT ) , problem 1 : part c . For the differential equation , $\frac{dy}{dx} = y^2 - x^2 $ , There exists a number $y_o$ such that if $y$ is a solution with ...
3
votes
1answer
119 views

Ordinary Differential Equation question

I'm stuck with the following ODE : $$y(4x+3y^3)\,dx+x(2x+5y^3)\,dy=0$$ It's not exact and the integrating factor is neither a function of $x$ nor a function of $y$ alone. Need some advice on how to ...
10
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1answer
323 views

Solving the continuous mice problem

The classic Mice Problem considers 4 mice standing on the four vertices of a square, and at some point every mouse starts running after its clockwise neighbour in a clockwise manner. It is not too ...
3
votes
2answers
122 views

Can I conclude that, if $v=0$ somewhere, then $v=0$ everywhere?

In solving a homework problem, I have encountered the following DE ($v$ is a function of $x$). $$v' = \frac{1}{x}\frac{1 + x^2}{1-x^2} v, \quad x \in (0,1)$$ I'd like to split the problem into two ...
3
votes
2answers
933 views

Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...
1
vote
2answers
116 views

Show that the solution to $\ddot x + \sin(x) = 0$ exists globally

Trying to show the solution to the equation in the title exists globally. I believe we are supposed to used "a priori" knowledge to show this rather than a rigorous proof. To get started, I multiplied ...
3
votes
0answers
180 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
2
votes
0answers
58 views

Solving an eigenvalue problem on the open unit rectangle

Let $\Omega=(0,1)\times(0,1)$ and consider the boundary value problem $$\begin{cases}\Delta^2u=f\\ u(x,y)=\Delta u(x,y)=0,& x,y\in\partial\Omega \end{cases}$$ I want to solve this boundary value ...
4
votes
3answers
267 views

How to solve vector-valued first order linear pde?

Is there an analytical solution to the pde system? $$\frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} = 0$$ $$\frac{\partial f}{\partial y} - \frac{\partial g}{\partial x} = 0$$ More ...
0
votes
4answers
136 views

Finding a differential equation when a half life is known

Does anyone know how I would write a differential equation for the following? I am not interested in the answer as such, I'm more interested in the steps and how to obtain the answer. I don't know how ...
0
votes
1answer
41 views

Comparing two DE problems and comparing how to solve them

So let's say use the function $y=Ax+B$ to solve $y'=sin(y-x)$ This I know how to do, as $A=1$ and $B=\frac{π}{2}$. But what if the problem was $y'=1.001sin(y-x)$? I have a feeling that the 1.001 ...
1
vote
2answers
93 views

Verifying a solution to a Differential Equation 2

How can I verify if this function is a solution of the differential equation? $ y' - 2ty = 1;$ $y = e^{t^2}\int_{0}^te^{-s^2}ds+ e^{t^2}$ I'm stuck, so any tip will be helpful Thanks in advance