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

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Solution $\overrightarrow {v}(t)$'s uniqueness & convexity of $\ln(\|\overrightarrow {v(t)}\|^2)$

I am lost and don't know how to prove the following: If $M$ is a positive definite symmetric square matrix and if $\overrightarrow {v}(t)$ is a solution of: $$\overrightarrow {v'}(t) = ...
4
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1answer
21 views

ODE solution - positivity and uniqueness

I always see the theorem on the textbooks, but they only state the theorem and then give examples on finding the roots of the equation. However, I would like to learn how it is proved that: If ...
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1answer
16 views

$e^{mx}$ in solving second order differential equations

In a book I am reading on differential equations, the author writes about the solution to a homogenous, linear, second order differential equation with constant coefficients. The author says something ...
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1answer
15 views

For what values of a,b this function satisfies the following differential equation and initial value

If i have this function: $$ x(t)= a\cdot e^t + b\cdot e^{-2\cdot t} $$ (1). For what values of $a$ and $b$ this function satisfies the following differential equation: $$ \frac{dx}{dt}+2\cdot x = ...
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1answer
23 views

differential equation - rectilinear movement of a boat using propulsion

I have a problem in my differential equation book that I can't solve because it gives me data that I can't seem to fit in the equation that the book gives me. This is something that I just don't get. ...
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1answer
27 views

What is the general solution of $2xydx + (x^2 + 2y)dy = 0$?

I need to check to see if the given $x^2y + y^2 = C$ is a general solution of the differential equation below: $$ 2xydx + (x^2 + 2y)dy = 0 $$ I eventually solved for $C$ and came to this conclusion: ...
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0answers
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Common factor in series solutions

a few times during my lectures on series solutions ( ODE) the teacher mentioned that it was only valid to use all the theorems and methods and such on analytic polynomials if they did not have a ...
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22 views

Show that the solution of this differential equation is analytic

Let $\alpha,\beta,a,b$ be real constants. Show that the differential equation given by: $y''= ay' + by \\ y(0)=\alpha\\ y'(0)=\beta$ has just one solution which is analytic in $R$ Solution: I am ...
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Related to degree of differentiat equation [on hold]

What is the degree of differential equation of family of ellipse having same foci ?
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1answer
41 views

Solution of differential equation - We find only one

I want to find all the solutions of the form $y(x)=x^m \sum_{n=0}^{\infty} a_n x^n, x>0 (m \in \mathbb{R})$ of the differential equation $x^2 y''+ xy'+x^2y=0$. I have tried the following: Since ...
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1answer
24 views

Escaping a stampede in Buffalo Country

A person is standing at a random point in square ABCD (vertices labelled clockwise) of side length 10 units. The person is capable of instantly reaching and running at 1 unit/second. A herd of ...
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1answer
14 views

operator method? non-autonomous differential equation.

Professor told me that solving: $\ddot{\theta}+\dot{\theta} +f(\theta) = c_0 \times \delta(t)$ can be done by taking a limit $\lim_{e \to 0} \int_{-e}^{e} \left ( \ddot{\theta}+\dot{\theta} ...
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How can I resolve: $ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $ via numerical solution?

How can I resolve a second-order ODE via Euler method? By example in the next ODE: $$ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $$ I know Euler method: $x_{i+1} = x_{i} + ...
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1answer
19 views

solving second order non-homogeneous differential equation 4

please help me to answer this differential equation : $ y''-2y'+2y= \cos(t) $ $ y(0)=1,y'(0)=0 $ I tried to solve this by assuming $r^{2}-2r+2=0$ but it ended up to minus $\Delta$ which wasn't ...
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0answers
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I need to solve the Mathieu equation: $y''(x)+(a−2q \cos(2x))y(x)=0$ [on hold]

I want to know what will be the boundary conditions? For integer order and $\pi$ periodicity? How I can find solution of equation?
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5answers
93 views

Derivation of the Exponential Nature of $e^x$

Presumably, the transcendental number $e$ was first found by taking the power series solution to the (arguably most fundamental) differential equation $f'(x)=f(x)$, with the initial condition $f(0)=1$ ...
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1answer
23 views

Differential equation with integration factor 2

please help me to solve this differential equation by integration factor : $ (3y-xy+2)dx+xdy=0 $ I tried to solve this and got $x^2 -x$ as integration factor but when I affected that on main ...
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0answers
29 views

clarification of a doubt over a defined result in ODE

I was going through the topic of Wronskian in ODE came up with the following result: I have a little doubt. Can we say the same if we interchange $y_1$ and $y_2$ i.e. between consecutive zeroes of ...
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2answers
29 views

Find value of $xy\sqrt{y^2 - x^2}$ for the given differential equation.

If $(y^3 - 2x^2y)dx + (2xy^2 - x^3)dy = 0 $ , then prove that the value of $xy\sqrt{y^2 - x^2}$ is a constant. This is what I've tried : $$ y(y^2 - 2x^2) dx + x(2y^2 - x^2) dy = 0 \\ ...
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1answer
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Use of Laplace transform to solve initial value problem.

--Short Explanation: I have to say I am going crazy with this problem as it does not give me the same as the suggested solution in the book: Problem: $y''-7y'+10y=9\cos{t}+7\sin{t}$ $y(0)=5$, ...
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1answer
23 views

Looking for help in regard to Series solutions with ordinary points (ODE)

I have a question that is in regard to the final answer that one is to get when solving some ODE questions via series. I am having some confusion on what if I am doing is correct/ why it is or is not ...
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1answer
15 views

Third-order differential equation with initial values using Euler method

The problem I have is the initial value problem $$y''' = x + y$$ with $$ y(1) = 3, y'(1) = 2, y''(1) = 1$$ that should be solved with Eulers method using the step length, $h = \frac{1}{2}$. The ...
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1answer
17 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
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1answer
36 views

solving second order non-homogeneous differential equation 3

Help me to solve this non-homogeneous differential equation : $ y''+y=\tan x $ $ 0<x<\dfrac{\pi}{2} $ I could reach to $y_{c}=c_{1}\cos x + c_{2}\sin x$ but particular solution is where I ...
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2answers
33 views

Differentiate $\sum_j\sum_i a(i,j)b(i,j)$ wrt $a(i,j)$

How do you differentiate a sum of a variable, wrt that variable, e.g. Find $\frac{dc}{da(i,j)}$ where $c = \sum_j\sum_i a(i,j)b(i,j)$. Context: I'm trying to find the jacobian.
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2answers
38 views

Solving differential equation $x'=\frac{x+2t}{x-t}$

I am trying to solve the following differential equation: $$x'=\frac{x+2t}{x-t}$$ with initial value condition: $x(1)=2$ This is what I have so far: Substitution: $u=\frac{x}{t}$ $$\implies ...
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1answer
36 views

Differentiate this power series

I am working on a problem which involves the differentiation of a power series. I know that that the following holds. Let $R$ be the radius of convergence of the power series $\sum_{n = 0}^\infty ...
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2answers
46 views

second order differential equation 1

I tried to solve this equation and reached to below answer but I think it needs to some recheck.please help me to get sure of this differential equation : $ 2y^{2}y''+2y(y')^{2}=1 $ my guess for this ...
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2answers
56 views

solving differential equation 1 [on hold]

please help me to solve this differential equation : $$ y'= \frac{x^2+3y^2}{2xy} $$ I found this as answer but I'm not sure : $$ \ln|x|=\ln\left| \frac{y^2}{x^2}+1\right|+c $$
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2answers
30 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
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2answers
73 views

Solving differential equation $x''(t)=x^6$.

Solve the following differential equation $$x''(t)=x^6(t)$$ If I had $x'(t)$ instead of $x''(t)$ the exercise would have been easier for me. I would appreciate some help with this problem. Thank ...
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1answer
11 views

Synchronization of Rossler system - the Rossler Attractor

I am studying synchronization of Rossler system given by the following set of two linear ODEs and one nonlinear ODE: $\dot{x_1} = -x_2 - x_3$ $\dot{x_2} = x_1 + ax_2$ $\dot{x_3} = c + x_3(x_1 - b)$ ...
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Elliptic Partial differential equation [on hold]

I could not understand the proof of Theorem 8.24 in Gilberg Trudinger's book. They Stated that it follows from Theorems 8.17 and 8.22. One more thing that how can i apply Lemma 8.23 to conclude the ...
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2answers
18 views

How to show that the general solution has the form:

We have an ode as such: $y'' + (sint)y'+t^2y=0$ Also, we know that $y_1$ and $y_2$ are linearly independent solutions. How to show that the general solution has the form: $y=c_1y_1+c_2y_2$ where ...
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2answers
40 views

Particular solution of the differential equation $ y' + (2/3)y = 1-(1/2)t, y(0)= y_0 $

I have this particular differential equation: $$ y'+(2/3)y = 1 - 1/2t, y(0) = y_0$$ I have to find the specific value $y_0$ where the solution touches t axis, but it does not cross it. I found the ...
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0answers
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Differential equations??? [on hold]

Two tanks of salt solution are connected to one another, with Tank 1 containing 30 gal of water and 25 g of salt and Tank 2 containing 20 gal of water and 15 oz of salt. Water with 1 g/gal of salt ...
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1answer
23 views

Use of undetermined coefficients issue

I'm given the problem $$y'' + 4y' = t$$ and asked to solve for y. I compute the general solution (using the characteristic equation) to be $$c_1 + c_2e^{-4t}\ ,$$ which I am pretty sure is correct. ...
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1answer
33 views

Solve the system of differential equations

I plan on adding more into later just a bit stuck, researching it at the moment. Solve the system of differential equations $$\begin{bmatrix} x'\\y' \end{bmatrix} - \begin{bmatrix} -11&15\\ ...
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2answers
29 views

Second order differential equations where rhs $= 6e^2\cos(3x)$

Solve the differrential equation $$y'' - 4y' + 13y' = 6e^{2x}\cos(3x)$$ where $y(0)=3$ and $y'(0)=-8$ I think we start like... For the homogenous case $$\lambda^2 -4\lambda + 13 = 0 $$ ...
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2answers
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Finding particular solution to inhomogeneous system of differential equations

I am asked to find the general solution set of the following system of differential equations: $$\begin{cases} x' = 3x -2y-2 \\ y' = 6x-4y-1 \end{cases} $$ I found the general solution set of the ...
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4answers
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First order differential equation: did i solve this equation right

So i'm trying to solve: $$x^2\frac{dy}{dx} + 2xy = y^3$$ I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ I think i've solved it and ...
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0answers
33 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
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0answers
22 views

Newton backward and forward interpolation (for ODEs) intuition.

For Newton's backward and forward formulas, I understand everything algebraically, but can someone please explain me this formula intuitively, especially intuition how "powers of the forward ...
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0answers
8 views

Homogeneous and Nonhomogeneous ODEs - where the name comes from?

Why differential equations can be called Homogeneous and Nonhomogeneous? I understand equations behind these names, but where the word "homogeneous" comes from?
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33 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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1answer
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Trying to use the method “Stiff” (Rosenbrock method implementation) from the book “Numerical Recipes in C”.

The program is compilable but I don't think it works correctly. According to the book, we need also method "odeint" for adaptive stepsize adjustment and fully implement Rosenbrock method. I used the ...
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Solving Differential equation using Frobenius Method [on hold]

I want to solve a differential equation using the Frobenius method but unable to do it.Please anyone solve this for me.The equation is $$x(1+x)y''+3xy'+y=0$$
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1answer
37 views

First order differential equation: how do I prove that $u$ satisfies the differential equation

So I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ now it says: Show that if $y$ is the solution of the above Bernoulli differential ...
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0answers
20 views

Applied mathematics for Clinical Medicine [on hold]

I'm a medical graduate, looking for advice/help on a project I would like to start. I would like to use applied mathematics to deconstruct the medical SOAP note into data sets that can be reproduced ...
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1answer
14 views

Need help with Laplace transform of piecewise /step functions

Hi I am having trouble figuring out how to calculate the laplace transform for $f(t)$ where $$f(t)= \begin{cases} e^{4t} & \text{if $ 0 \lt t \lt 2 $} \\ 1 & \text{if $ t \gt 2 $} ...