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

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uniqueness of complete integral

Is the first order nonlinear PDE that the given complete integral solves unique? I had this question while solving the problem 3-3 of "Partial Differential Equations" written by Evans.
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2answers
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2nd order nonlinear ODE question

I am looking for help to solve the following $F(x,y(x),y'(x),y''(x))=0$ equation: $$ xy''(x)-y'(x)-(x^2)y(x)y'(x)=0 $$ Very much appreciated.
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1answer
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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: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
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1answer
24 views

Proving this Differential Equation has one solution

Suppose q(x) is an n-th degree polynomial and consider the following DE: x(dy/dx) + y = q(x) Show that there is only one solution to this differential equation that is, itself, a polynomial. What is ...
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Explicit solution to a first order nonlinear ODE

Is there any explicit solution to the following ODE? $G'(z) =aG(z)+bG(z)^α-c$ $G(0) = d_0 $
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2answers
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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 ...
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2answers
107 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, ...
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1answer
25 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 ...
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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 $\mathbb{R}$. ...
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1answer
17 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 ...
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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 ...
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Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
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1answer
61 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. ...
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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 ...
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1answer
30 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 ...
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1answer
21 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 ...
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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 ...
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0answers
14 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 ...
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2answers
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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 ...
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1answer
22 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 ...
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1answer
20 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!
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1answer
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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 ...
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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 ...
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1answer
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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 ...
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3answers
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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 ...
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1answer
25 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 ...
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0answers
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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 ...
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1answer
17 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 ...
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3answers
50 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 $(x,y)=(0,0)$ } ...
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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 ...
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1answer
162 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 ...
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0answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\phi$ ...
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4answers
58 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 ...
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2answers
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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 ...
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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 ...
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2answers
39 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. ...
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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 ...
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45 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 ...
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1answer
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Find the solution of the Differential equation [on hold]

Solve $$\frac{x+y\frac{dy}{dx}}{x\frac{dy}{dx}-y}=\frac{\left( 1-\left(x^2+y^2\right)\right)^{\frac{1}{2}}}{\left(x^2+y^2\right)^{\frac{1}{2}}}.$$
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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] + ...
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2answers
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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 ...
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2answers
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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 ...
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1answer
87 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 ...
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Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
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1answer
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 ...
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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, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...