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

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

Asymptotic behaviour of $\varphi''(x)=F(\varphi(x))$

I'm concerned with the discussion of a ODE, especially the discussion of the solution. I've got the assumptions that there is the relation $\varphi''(x)=F(\varphi(x))$ for all $x$ on $\mathbb{R}$. ...
0
votes
1answer
16 views

Differential equation with no nontrivial periodic solution

We are given $f=(f_1,f_2): \mathbb{R}^2 \rightarrow \mathbb{R}^2$, $C^1$ class with the property: $$(1) \ \ \ \forall_{(x,y)\in\mathbb{R}^2} \frac{\partial f_1}{\partial x}+\frac{\partial ...
0
votes
1answer
15 views

Can I extend these ODE formulas to complex numbers?

In my calculus class, we recently covered first-order, linear ODEs. Specifically, we discussed the formula for the solution of one (and its derivation): $$y=\frac{1}{u(x)}\int Q(x)u(x)dx$$ where ...
0
votes
1answer
29 views

Inviscid Shallow Water Equation

Aside from wikipedia where might I find a fairly comprehensive, yet simple to read, piece of literature on the inviscid shallow water equation? Can you recommend any texts? I don't want literature ...
0
votes
0answers
26 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
1
vote
1answer
50 views

Looking for a way to solve this differential equation.

Can somebody give me a hint how to try to solve the following differential equation: $ \ddot{r} - \frac{1}{r^3} = 1$ where $r = r(t)$ and $\ddot{r}$ is the second derivative. It is not homework btw. ...
0
votes
2answers
3k views

Differential Equations; Mixture problem

A textbook example asks me: A large tank is filled to capacity with 100 gallons of pure water. Brine containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gal/min. The ...
0
votes
1answer
26 views

Prove that $\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth(\frac{\pi}{2})$

I want to prove that: $$\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth\left(\frac{\pi}{2}\right)$$ Using the fourier series: $$\phi(x) = \begin{cases}0 & \text{if ...
0
votes
1answer
15 views

How to solve this two variable Bernoulli equation ODE?

I'm trying to solve this homework question but the two variables is throwing me off. Which one is my standard $t$? How do I handle both variables? I'm to solve this Bernoulli equation via substitution ...
1
vote
1answer
22 views

how to write a differential equation for a problem like this

I've got a problem and i should solve it using differential equation.I don't know how to write the equation and start. A person is trying to fill a bathtub with water. Water is flowing into the ...
0
votes
0answers
13 views

homogeneous BVP has at most one linearly independent solution

I am trying to understand following proof. I understand the set up however can't make the connection with the Picard Lindelöf Theorem. Can you please help me with this? Statement: The homogeneous ...
1
vote
2answers
43 views

Reachable Space by an ODE

Let $\dot{x}(t) = Ax(t) + Bu(t)$ be an $n$-dimensional first order ODE where $u(t) \in \mathcal{P}$ for some convex polytope $\mathcal{P}$, for every $t \in \mathbb{R}$. Assume $x(0) = 0$. Is there a ...
0
votes
1answer
20 views

Solution of the heat equation

Let $u:\mathbb{R}^n\times(0,+\infty)\to\mathbb{R}$ solves the following heat equation: $$u_t(x,t)-\triangle u=0,\quad (x,t)\in\mathbb{R}\times(0,+\infty)$$ (a) Show that for each ...
0
votes
1answer
19 views

is this differentiable form exact?

Take $\omega^{2}$ a 2-exact form and $\omega^{3}$ a 3-closed form, the question is, can we have that $\omega^{2}\wedge\omega^{3}$ be exact? Thanks!
0
votes
1answer
21 views

What can I think of the function $F$ that's being used for most(?) explicit first order ODEs?

Almost anything on this topic only deals with how to solve ODEs, but so far I couldn't find one single site defining this ominous $F$ that's being used so often, like in Wikipedia or in my script (not ...
1
vote
0answers
23 views

Differential Equation Find general solution of y'' - y =cosh(x) using variation of parameters

Hello I am having some issues with the simplification of the DE, I am okay up on till $$y_p(x)=v_1(x)y_{p1}(x) + v_2(x)y_{p2}(x) $$ $$ \frac12 e^{-x}\left(\frac {-e^{2x}}4-\frac x2\right)+ \frac ...
0
votes
1answer
57 views

How to solve this system of 3 ODE?

I would like to know how to solve this system of differential equation. It consist of 3 ODEs, describing the behavior of an Induction Machine supplied with DC voltage. I a interested to derive the ...
5
votes
3answers
3k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
0
votes
0answers
21 views

How to determine behaviour of this derivative in the following differential equation?

Given the following differential equation $$\frac {dx}{dt} = ax + \cos(x)$$ for some $a \in \mathbb R$. I need to determine the shape of the direction field of $x(t)$ in the vertical axis and $t$ in ...
1
vote
1answer
24 views

Compound interest Differential Equation

A college student starts a savings account with an initial balance of $\$0$. He plans to save money at a continuous rate of $\$200$ per week. Also, at every week he plans to increase this rate by ...
0
votes
0answers
7 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
0
votes
1answer
16 views

System of linear differential equations - generalized eigenspaces

I'm trying to prove that if $f$ is a solution of the system $x'=Ax$ (where $A \in M_{n}(\Bbb R)$) such that $f(0)=x_0 \in G_\lambda$ ($G_\lambda $ is the generalized eigenspace for $\lambda$) then ...
1
vote
3answers
47 views

The function $4x^3y/(x^4+y^2)$ fails the Lipschitz condition near the origin

I have to prove that Lipschitz condition is not satisfied for the function, $$ f(x) = \begin{cases} {4x^3y \over x^4 +y^2}, & \text{if $(x,y) \neq (0,0)$ } \\ 0, & \text{if $(x,y)=(0,0)$ } ...
0
votes
0answers
7 views

Differential Algebraic eqn using Adomian Decomposition Method [on hold]

Refer to research paper http://www.gbspublisher.com/ijpamsv3/ijpamsv3n1_10.pdf, in example 1 i am unable to understand how author has calculated u1,0 =14xsinx+sinx-xcosx. Can somebody plz ...
3
votes
1answer
158 views

On definition of gamma function.

We all know that gamma function's definition is $$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$ and it is divergent for $x<0$. Yesterday, I was studying about Bessel ...
2
votes
0answers
23 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
0
votes
0answers
8 views

system of ordinary differential eqs.

I want to solve the following system of ordinary differential equations: $F_1'(t)=-i \lambda \sqrt{n+1}F_2 e^{i \Delta t} $ $F_2'(t)=-i \lambda \sqrt{n+1}F_1 e^{-i \Delta t} $ All paratmeters ...
2
votes
3answers
45 views

Second-order inhomogeneous differential equation $y''\:-\:4y'\:+\:2y\:=\:2x^2$

I'm trying to solving a 2nd-order inhomogeneous differential equation, but I'm not sure with my answer since I'm only learned it by myself & it has nothing to do with school or homework, so I have ...
0
votes
0answers
25 views

Maxwell equations in vector notation

Our professor has given us the following assignment - "Maxwell equations in vector notation" - and having not studied them, I looked them up yesterday. The equations were pretty clear, divergence and ...
1
vote
1answer
30 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: (1+2y^2)(dy/dx) + (3-y)cosx = 0 I ruled out the methods I've so far learned in class ...
3
votes
2answers
35 views

Eigenvalues of Differential Equation with Boundary Condition

Here is a problem from my homework assignment that I am struggling with: Consider the differential equation $\frac{d^2\phi}{dx^2}+\lambda\phi=0 $. Determine the eigenvalues $\lambda$ if $\phi$ ...
1
vote
4answers
57 views

How to solve $\ddot{x} = x + 8e^{3t}$ without Laplace transform?

How do you solve this diff-eq without using laplace transforms? $\ddot{x} = x + 8e^{3t}$ That $8e^{3t}$ is throwing me off...Also, I need to get two constants in the answer so I can solve for ...
0
votes
2answers
30 views

How does the book arrive to the solution cos^2(x)*cos(2y) using separation of variables method?

Problem: sin(x)*cos(2y)dx+cos(x)*sin(2y)dy=0, y(0)=pi/2 These are my steps: sin(x)*cos(2y)dx+cos(x)*sin(2y)dy=0 (sin(x)/cos(x))dx+(sin(2y)/cos(2y))dy=0 tan(x)dx+tan(2y)dy=0 ...
0
votes
1answer
31 views

Using eigenvalues to determine stability

One needs to show why solutions for the system $$x'=\left[\begin{matrix} 0&-1&0 \\ 0&-2&0 \\ -1&2&-1\end{matrix}\right]x$$ are Lypunov or asymptotically stable/unstable. Most ...
1
vote
2answers
37 views

Population dynamics calculation

I am trying to solve the question: A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero, the population consists of two members. ...
0
votes
3answers
32 views

integrating factors

Hello! I have been working on some differential equation homework in preparation for an upcoming exam. I understand that when trying to solve a differential equation that is not exact sometimes an ...
0
votes
3answers
43 views

Linear nonhomogenous ODE

Solve: $$y''(t)=-y(t)-\cos(t),\ \ \ \ \ \ y(0)=y'(0)=0$$ I'm sorry that I have no approach to solve it. In my solution I have the hint to solve it with resonance but I don't know anything ...
1
vote
1answer
42 views

Find the solution of the Differential equation [on hold]

Solve $$\frac{x+y\frac{dy}{dx}}{x\frac{dy}{dx}-y}=\frac{\left( 1-\left(x^2+y^2\right)\right)^{\frac{1}{2}}}{\left(x^2+y^2\right)^{\frac{1}{2}}}.$$
1
vote
1answer
50 views

Techniques to solve such a PDE

I have the eigenvalues problem on $[0,\pi] \times [0,2\pi]$ $$\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \left[\sin\theta \frac{\partial}{\partial \theta}\right] + ...
1
vote
2answers
36 views

How to solve this coupled linear differential equations?

$\partial_t f(x,t)= \alpha \partial_x^2f+\beta f + \gamma g \\ \partial_t g(x,t)= \alpha \partial_x^2g -\beta f - \gamma g$ With everything real. I tried to take the first equation and express ...
2
votes
2answers
52 views

Solution of differential equation with some conditions!

Find twice differentiable function $f:\Bbb{R}\to \Bbb{R}$ such that $f''(x)=(x^2-1)f(x)$ with $f(0)=1$ and $f'(0)=0$ I can see that $f(x)$=$e^{-x^2/2}$ satisfies the required conditions but I don't ...
1
vote
1answer
86 views

How to solve $y'=\frac {e^{x-y}}{y-1}$?

Could you please give me some hint how to solve this problem. Suppose that $y(x)$ solves the differential equation $y'=\dfrac {e^{x-y}}{y-1}$ and that $y(x)\to 0$ when $x\to -\infty$. Compute ...
2
votes
0answers
45 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
4
votes
2answers
51 views

How to solve $xy=2\int_1^xy(t)dt+5$?

Could you please give me some hint how to solve this equation: $xy=2\int_1^xy(t)dt+5$. It is not known whether $y(x)$ is continuous or not, so I could not use Fundamental Theorem of Calculus for ...
1
vote
0answers
62 views
+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
2
votes
2answers
41 views

Please explain the Quotient Rule

I am currently working on an equation but I'm having a hard time understanding how to get the answer. the answer is ${(x^2-4)(x^2+4)(2x+8)-(x^2+8x-4)(4x^3)\over (x^2-4)^2(x^2+4)^2}$ The equation is ...
0
votes
0answers
29 views

Fourier series of $\phi(t) =0$ if $ -\pi \lt t < 0$ and $\phi(t)=\sin(t)$ if $0 \le t \lt \pi$ [duplicate]

I am having trouble finding the Fourier series of: $$\phi(t) = \begin{cases}0 & -\pi \lt t < 0 \\ \sin(t) & 0 \le t \lt \pi \\ \end{cases}$$ I am trying to use: $$\phi(t) = ...
0
votes
1answer
15 views

Solve the differential equation using power series.

$\displaystyle y^{'} = {\frac{y}{x}} + 1$ cannot be solved for $y$ as a power series $x$. Solve this equation for $y$ as a power series in powers for $ x-1 $.: Introduce $t=x-1$ as a new independent ...
2
votes
1answer
58 views

rough question in Differential Equation.

I'm trying to solve the following system of differential equations, but I couldn't find any method / procedure to obtain the solution. I don't want a comprehensive and complete answer; a hint will ...
0
votes
2answers
20 views

Application in linear differential equations

Hey I'm a little stuck on where to proceed on this question: The initial value problem: $$x′(t) = ae^{−bt}x(t)$$ $$x(0) = x_0$$ arises from a model of tumor growth. The constants a and b are ...