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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1answer
15 views

Existance and uniqueness of an ODE

I have the following ODE: $y'=y+e^{-2t}y^2$ I know the solution is $y=\frac{1}{e^{-2t}+ce^{-t}}$, c constant. Then the problem says that $y=0$ is a different solution. How can I explain that this ...
2
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1answer
19 views

Rewrite a differential equation formula

I am currently reading Elementary Differential Equations and I don't quite understand how they rewrote this Differential equation. I know that it is a very simple answer, but for some reason I can't ...
0
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0answers
14 views

The difference between euler line and direction field?

Can someone please explain to me what the difference is between the direction field and euler lines? Direction field is the tangent line to the integral curve of a differential equation, as far as I ...
0
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0answers
11 views

Finding solutions for a nonlinear ODE of the second order

Try to solve in closed form the ODE (any kind of special functions can be used, or a solution in parametric form is also valid) yy'' + y' = x, (1) y(x_0) and y'(x_0) are open for any x_0. Note ...
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0answers
13 views

Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
2
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2answers
17 views

If $f(x) \cdot x < 0$ for all $x \in \partial B_R(0)$, then the IVP $x' = f(x)$, $x(0) = x_0$ has a global solution.

I have a homework problem that asks If $f : \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and satisfies $$ f(x) \cdot x < 0 \quad \quad \text{for all } x \in \partial B_R(0) ...
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0answers
12 views

Generalizing Method of Characteristics: Example

I have consulted other sources, but the relevant ones have used notation that I am not entirely comfortable with. I'm told that generalizing this procedure is straightforward, but I find myself ...
7
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1answer
112 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
1
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3answers
33 views

Differential equation using substitution

I have the following differential equation with a hint to solve of using the substitution of $u = \frac{y}{x}$: $$\frac{dy}{dx} = \frac{(y/x)^3 + 1}{(y/x)^2}$$ I was wondering, after the ...
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2answers
34 views

Solutions of a second order nonlinear differential equation

I am in trouble solving the following differential equation: $$\ddot{x}(t)=-\alpha\dot{x}(t)\dfrac{x(t)}{\left(\beta^2+x(t)^2\right)^{\frac{5}{2}}}$$ where $\alpha$ and $\beta$ are constant. How can I ...
0
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2answers
33 views

First order differential equation with time-varying parameter

If I have: $\dot{\sigma}(t) = -\gamma \sigma(t)$ where $\gamma$ is a constant, the solution is given by: $\sigma(t) = \sigma(0) e^{-\gamma t}$ Now what if I have the differential equation: ...
0
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0answers
24 views

find second order differential equation

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ My question is how to find $f$? Can show some steps? Thanks for help.
0
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1answer
19 views

seperation of variables difficult integral

$x'(t)=x(t)*(a-x(t)), x(0)=x_0$ I have to solve this ODE with seperation of variables. But I have problems. $\frac{\partial x}{\partial t}=x(t)*(a-x(t)) \iff ...
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0answers
6 views

Sturm-Liouville: Prüfer's substitution and convexity (concavity) of the phase function

Consider the following system with parameter $\lambda$ (we don't fix the right-hand side condition): $$ \left\{\begin{align} & -\psi''(x,\lambda) + q(x) \psi(x,\lambda) = \lambda ...
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0answers
28 views

Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature ...
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3answers
29 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
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0answers
22 views

Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
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0answers
11 views

uniqueness of complete integral [on hold]

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.
3
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2answers
62 views

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.
0
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1answer
30 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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0answers
27 views

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 $
5
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2answers
115 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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0answers
20 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
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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 ...
0
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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 ...
1
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1answer
23 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
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0answers
15 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 ...
0
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0answers
29 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
0
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1answer
28 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 ...
0
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1answer
22 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
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1answer
24 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 ...
1
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0answers
27 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
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1answer
35 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
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2answers
38 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
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1answer
70 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
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1answer
22 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 ...
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0answers
10 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 ...
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0answers
9 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
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1answer
163 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
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3answers
62 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
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1answer
18 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 ...
2
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0answers
27 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
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1answer
15 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 ...
0
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0answers
28 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
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1answer
27 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 ...
3
votes
1answer
44 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: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
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4answers
61 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
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2answers
32 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 ...
1
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2answers
41 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
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3answers
51 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 ...