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

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Solving the following differential equations

How do we solve the following differential equation: $$\frac{dy}{dx}+ yx^2 = \frac{7x^2}{y}$$ I was hoping to use integrating factors, however that seems inapplicable as the y term is present on ...
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0answers
42 views

Laplace Transform for solve ODE (RLC circuit)

I have an RLC circuit and I want to know the charge on the capacitor $q(t)$ using Laplace transform: The diferential equation is: $$ Lq'' + Rq' + \frac{1}{C}q = E(t),$$ where $L = 1H , R = 20 ...
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49 views

solving a 2nd order PDE with constant coefficients

This question is followed up from this question system of non-homogeneous advection equations \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ ...
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0answers
28 views

Why do Runge Kutta's method and Euler's are so different?

I am solving a $\underline {\dot A}=\underline A\cdot \underline x$ system of linear equations numerically. I have don'e this in the popular of methods of Euler and Runge Kutta. I have noticed a ...
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1answer
25 views

Analytic solution to the one-compartment model

I have the following linear system of ordinary differential equations: $$\frac {dA} {dt} = -k_a \cdot A$$ $$\frac {dC} {dt} = k_a \cdot A - k_0 \cdot C$$ $$A(0) = A_0$$ $$C(0) = 0$$ Some people may ...
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0answers
26 views

Effect of adding a constant to the torsion of a 3D curve

Let $\gamma$ be an arc-length parametrized curve in $\mathbb{R}^3$. Let say I add a constant to the torsion of $\gamma$ and let $\widetilde{\gamma}$ be the curve associated to the curvature of ...
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0answers
33 views

Solving differential equation and the inequality

I have stuck in a small step were I need to solve for t: $$ e^{t(\lambda_3-\lambda_1)} \geq 1 \Rightarrow t \geq\frac{1}{\lambda_3 - \lambda_1}$$ I don´t understand how the solution is required. ...
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1answer
97 views

Kovacic's algorithm

Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm? Or can anybody help me to ...
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1answer
49 views

What exacty is the role played by Jacobian or Wronskian?

In many of our derivations or in differential equations we come across the terms Jacobian or Wronskian. For example, to check the linear independence of solutions of differential equations, we ensure ...
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68 views

Integration of a given Integral

Given the integral $$ \hat{\alpha}({r_{0}})=2\int^{\infty}_{{r_{0}}}\frac{dr}{r \sqrt{1-\frac{2M}{r}} \sqrt{\left(\frac{r}{{r_{0}}}\right)^{2}\left(1-\frac{2M}{{r_{0}}}\right) ...
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2answers
26 views

Using optimization for a logarithmic function

Question: A tangent line is drawn on the graph of $y=\ln x$ for $0\lt x\lt 1$. A right triangle is thus formed in the fourth quadrant. If we regard the area of this triangle has a positive value, ...
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2answers
44 views

Newton's Law of Cooling Example

A $200°F$ cup of tea is left in a $65°F$ room. At time $t=0$ the tea is cooling at $5°F$ per minute. Write an initial-value problem (differential equation with an initial condition) that models the ...
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2answers
67 views

Looking for a function that fits a certain criteria

I am looking for a function that fits this description: $$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$ or $$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$ For all values of $n$, with this function i am looking to ...
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4answers
46 views

Name/Solution of this Differential Equation

I have a differential equation of the form $$\frac{d^2y}{dx^2} + \omega \frac{dy}{dx} = 0$$ Where $\omega$ is a constant I was wondering what kind of solutions this differential equation has. Just ...
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1answer
24 views

Differential equations logarithmus rule question

I'm trying to understand a exercise about differential equations $x'=\frac{1}{2}x+1$ I'm going for the general solution by using separable equation. Everything goes well until I get off the rails: ...
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2answers
215 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
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1answer
25 views

Phase plane and chain rule

Background: In section 5.1 Linear systems in A First Course in Differential Equations, Second Edition by J. David Logan , there is an example, Example 5.4. Here you consider $ x'=2y\\ y'=x $ and ...
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4answers
70 views

Solving differential equation with inhomogeneous part $\sin x \cos x$

How do you solve the following inhomogeneous differential equation: $$ y' + y\cos x = \sin x \cos x ?$$ I determined the homogeneous solution ($y_h=Ce^{-\sin x})$, but how do I find the ...
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2answers
41 views

Differential Equations- Wronskian Fails?

I was doing a problem where the goal was to find whether two functions: f(x) = sin(2x) , and g(x) = cos(2x) are linearly independent or not using the wronskian. The problem is simple enough, and after ...
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0answers
26 views

Decidability - Complexity

Can someone tell me where I can get some information about the following? We have linear differential equations with polynomial coefficients depending on x. $a_n(x)y^{(n)}+ \dots ...
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0answers
54 views

system of non-homogeneous advection equations

I would like to solve this system \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ \end{array} \right. \end{equation} First , I would like to ...
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1answer
61 views

How does one solve this second order non-linear differential equation?

$$y \, y'' + \frac12 (y')^2 - y' = 0$$ I can't figure out which approach I should use to solve this ... anyone got a hint?
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1answer
34 views

What method should I use to solve this differential equation? [closed]

$$ y'-y^2+8y-15=0,~y(0)=4 $$ I used the Bernoulli method, but couldn't find the right answer.
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65 views

Poincaré and Liapunov Theorem

I'm reading this paper, where they claim the equation \begin{equation} u'' + c u' + u(1-u) = 0 \end{equation} is linearised about $u = 0$. The equation \begin{equation} u'' + cu' + u = 0 ...
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1answer
27 views

I need to understand, how the P in the $bP$ is equal to $K(1-(b/a)$ in the following

From An Invitation to Biomathematics: To illustrate this concept mathematically, assume that a population grows according to the logist model $\frac{dp}{dt}=a(1-\frac{P}{K})P$ and that harvesting ...
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1answer
20 views

Simplifying a nonhomogenous ODE via limits as t --> infinity.

I am self studying a systems biology paper on segmentation in evolutionary developmental biology and trying to replicate the simulation. The simulation implements systems of differential equations ...
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1answer
61 views

Solution to $y'=\frac{(x+y)^2}{(x+2)(y-2)}$

$$y'=\frac{dy}{dx}=\frac{(x+y)^2}{(x+2)(y-2)}$$ According to me it is not a homogenous equation as the degree of the terms is different. I even tried reducing it to some of the standard forms, but I ...
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2answers
50 views

Differentiate using the chain rule

Trying to get me head around the chain rule... Differentiate the following using the chain rule: $ln({x-1\over x^3})$ so $f(x)ln={1\over x}$ and $f(x){x-1\over x^3}=-{3x-1\over x^3}$ What's the ...
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2answers
34 views

Solve linear differential equation

So I have the following linear differential equation $$t\frac{dy}{dt}-3y=t^4$$ My first step was to divide through by $t$ to give $$\frac{dy}{dt}-3t^{-1}y=t^3$$ Then to find the integrating factor ...
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0answers
23 views

Examples of ODEs with a 3-dimensional vector-function

I'm testing a program which solves ODEs and I need some examples. Where can I find some examples of ODEs with a 3-dimensional vector-funcion? Likes these two:
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2answers
21 views

Looking to create a non-linear phase portrait of an “Elliptical Spiral”

I am looking to find an equation that will plot what i like to call an elliptical spiral. But i do not know where to begin. What i mean by an elliptical spiral is this: You take a point on the y-axis ...
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2answers
23 views

Differentiate exponential/logarithm function

Given the function, $e^{2x+1}\ln3x$, I can differentiate the two separately but how do I combine them? $\dfrac{d}{dx} e^{2x+1}$ = $2e^{2x+1}$ $\dfrac{d}{dx}\ln3x$ = $\ln3$
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0answers
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Why is it possible to use the “separation of variables” method for solving integrals? [duplicate]

Heyho, I have been using the separation of variables method for quite a while now, but what has always bothered me a bit is, why is it possible to do use this method? I'll give a concrete example ...
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7answers
772 views

Why does the “separation of variables” method for DEs work? [duplicate]

Heyho, I am using the separation-of-variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operations. I'll give a concrete example (source ...
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1answer
42 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in ...
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1answer
24 views

Maximum condition for second order differential operators

Let $A$ be a second order differential operator such that $$Af(x) = \sum_{ij} a_{ij}(x) \big(\partial_i \partial_j f(x)\big) + \sum_j b_j(x) \partial_j f(x) $$ Assume that $x \in B(0,r)\Rightarrow ...
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1answer
30 views

A second order linear ordinary differential equation

I am trying to solve the following differential equation: \begin{equation} (1-x^{2})y''+\{\alpha x+1 \}y'+\{\beta^{2}x^{2}-\beta{x}+\gamma\}y=0 \end{equation} where $y=y(x)$. $y'$ and $y''$ are the ...
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1answer
36 views

construct a path in $\Bbb{R}^n$ with specific derivatives

If we want a curve $\gamma:(-\epsilon,\epsilon)\to \Bbb{R}^n$ to have $\gamma(0) = x$, and $\gamma'(0) = b$. It suffices to take $\gamma(u)= x + u \cdot b$. At the same time, I have a function on ...
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0answers
20 views

comparison of norms in vector ODEs

Assume I have the following equation $$ \dot{f}(t) \le A(t)f(t),\quad f(0) = f_0$$ where $f:[0,\infty]\rightarrow\mathbb{R}^n$ has positive components ($\ge0$), ...
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1answer
18 views

Solving a certain type of differential equation of a higher order.

$$xx'''=1$$ then the substitution is made: $$x'=p(x)$$ then it says that :$$x''=p \frac{\partial p }{\partial x}$$ why is this , shouldn't it be just $$\partial p \over \partial x$$ What would ...
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1answer
22 views

Diffrential equation question, cannot find constants.

The rate of cooling a body is given by $\frac{dθ}{dt}=kθ$, where k is a constant. If $θ$ is 60°C when $t$ is 2 minutes and 50°C when $t$ is 5 minutes, determine the time taken for $θ$ to fall to ...
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0answers
130 views

Second order coupled differential equations

I am trying to solve two coupled ordinary differential equation. $x''+Ax'+By'+Cx+Dy=U$ ; $y''+Ex'+Fy'+Gx+Hy=V$ ; $A,B,C,D,E,F,G$ & $H$ are constants, $U,V,x$ and $y$ are function of time. All ...
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1answer
38 views

A question on Green's functions & integral operators

I'm fairly new to the concept of Green's functions, but from what I understand so far, they are a powerful tool for solving PDEs with boundary conditions. Given a differential equation (in operator ...
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2answers
30 views

Chain Rule and Multivariable calculus

I would be very grateful if anyone could assist with the following: Given that $$ z = yg(x^2-y^2) $$ I'm trying to show that: $$\frac{1}{x}\frac{\partial z }{\partial x} + \frac{1}{y}\frac{\partial ...
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2answers
37 views

Differential equation for finding closest point on surface.

Inspired by this question I got to think about a more general case. Say I have any discretized surface and want to find closest point from each point outside of surface to the surface. Say that I can ...
2
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2answers
70 views

Solution to system of linear ODE's

Let $\Delta_n$ be the closed unit simplex in $\mathbb R^n$. For any $a,b \in \Delta_n$, define the differential equation: $$ a'(u) = b-a(u) \quad\quad\quad a(0) = a $$ How does one go about solving ...
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0answers
18 views

Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered in origin. Thus the shortest distance between two points on the sphere is the geo-desic. Is there a transformation (linear or non-linear) on the points ...
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2answers
39 views

Find the Laplace inverse of the following.

$$ \frac{2s+5}{s^2+6s+34} $$ I am stuck on this part: Wolfram has the step by step showing that you simply split up the original fraction into $$ \frac{2s}{s^2+6s+34} + \frac{5}{s^2+6s+34} $$ and ...
2
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1answer
27 views

Proof of Liouville's formula , details and confusions. [Matrices, determinants..]

So I've got the homogeneous linear equation: $$x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ where $a_1(t)...a_n(t)$ are real continuous on intervals. This is what my textbook states: If ...
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2answers
39 views

Question about Differential Equations, why is there no constant for $x$?

The following equation is in the form$$\frac{dy}{dx}=f(y)$$ Solve the diffrential equation$$\frac{dy}{dx}=2+3y$$ $$\frac{dy}{dx}=2+3y$$ $$dy=(2+3y)dx$$ $$\frac{dy}{2+3y}=dx$$ then I ...