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

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1answer
15 views

Separable Equation Validity

When solving a "separable" ordinary differential equation, I've read that it's common practice to collect all terms involving the solution $y$ on one side, and all terms involving the independent ...
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1answer
10 views

system of 2nd order into 1st order

p'' = -q and q'' = p Goal: convert above system into first order equations. I expressed the system in matrix form. Next, did some calculation and ended up getting p' = e^t and q' = e^(-t) But I ...
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2answers
21 views

Homogenous differential equation of the first order

I cannot solve this question... excuse my bad english! Boiling water cools down proportionally to the temperature the water has at the specific point of time. What temperature does the water have ...
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0answers
17 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
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0answers
31 views

Does Every Periodic Function Have An Associated Differential Equation?

My question is the opposite to proving the existence of periodic solutions to ODE's. Assume that $\ f(z)$ is a periodic function over the $\mathbb{R}$ , or doubly periodic over some lattice $\Lambda$ ...
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0answers
20 views

the stability of ODE by studying the eigenvalues

I'm trying to study the stability of systems of odes but i fail to find the steps of positive and zero eigenvalues; and in the case of negative and zero eigenvalues. What is the procedure to follow ...
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0answers
20 views

Differentiation of Laplace Transform

It is known that The $s-$derivative rule states that $$ \mathcal{L} (t^{n} f) = (-1)^{n} F^{(n)} (s) $$ The proof for the laplace differentiation involves \begin{align*} F'(s) &= ...
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0answers
42 views

Matrix of a differential equation

I had recentely encounter my first exercise about merging matrices notions and differential equations functions, but after solving the differential equation, I don't know how to represent it in the ...
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0answers
20 views

Expressing a set of discrete inequalities as a continuous differential equation

I'm trying to work out the solution to a problem of sequential inequalities. I believe the solution collapses to a set of differential equations, but I'm having trouble organizing things and I think I ...
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1answer
22 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
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1answer
37 views

Confusion about Partial Derivative for a Function of One Variable

This question actually came up as I was reading an example in my differential equations book (Boyce & Diprima): Solve: $2x+y^2+2xyy'=0$ Define $\psi(x,y)=x^2+xy^2$ Then ...
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1answer
23 views

Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations: General form of ...
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1answer
18 views

Second Order Inhomogenous Differential Equation

I have run into an issue trying to solve this second order differential equation $ r''(t) - i r'(t) = -i\gamma[-\frac{1}{2} + \frac{1}{1+e^{-\alpha t}}], $ where $\alpha$ and $\gamma$ are real ...
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0answers
14 views

Prerequisites for some topics on two dimmensional ODEs

I am an Electrical Engineer student and I want to do an summer course on two dimmensional ODEs. The reference book is Arnold and the subjects that will be cover it's: Vector fields, fixed points, ...
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0answers
16 views

Runge-Kutta methods with strictly positive Butcher tableau

An explicit $s$-staged Runge-Kutta method for an autonomous ODE $\dot y = L(y)$ can be written as $$ k_i = L\left(y_n + \tau\sum_{j=1}^{i-1} a_{ij} k_j \right)\\ y_{n+1} = y_n + \tau\sum_{i=1}^s b_i ...
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1answer
44 views

Linear Systems: Exponentials of a Matrix

I have a rather odd question to some, but one that has stumped me for a good few minutes on a homework assignment that states: For each matrix, find the eigenvalues of $\text{exp}{(A)}$, ...
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1answer
18 views

Linear Systems and Linear Transformation

I want to confirm my attempt to see if I am on the right track. The question is as follows. Show that the operator norm of a inear transformation $T$ on $\mathbb{R}^n$ satisfies ...
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1answer
52 views

How do I numerically solve this type of differential equation? (Wave Equation)

I'm trying to solve the wave equation numerically. I'm brand new to this and what I'm basically trying to accomplish is simulating a plucked string with fixed endpoints. How do I find the $h(x,t)$ ...
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0answers
14 views

Relationship between Laplacian and Taylor expansion for 2nd derivative

I am working on converting a solution to a certain PDE from working on a regular 2D grid to work on a 3D triangular mesh. In the 2D scenario the 1st and 2nd derivatives are, of course, approximated ...
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1answer
17 views

How to differentiate between (x-absent) DE and constant coefficients DE?

x-absent second order differential equation is solved by the substitution ( $y'=u$ and $y''=u\frac{du}{dy}$ ). But this equation: $$y''+6y'+5y=0$$ can't be solved this way, it can be solved only ...
4
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1answer
29 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
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1answer
20 views

Solve DOE system with polar coordinates?

I am studying for a exam and one of model questions is solve a DOE system using polar coordinates. I've research and didn't find any reference about this subject. System in question is $$ ...
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2answers
49 views

Getting 0 solving Schrodinger equation with Dirac delta by Fourier transform

I am attempting to solve the Schrödinger equation with the potential $V = - \delta (x)$. This leads to a differential equation $$ \alpha \psi''(x) + (E + \delta(x)) \psi(x) = 0 $$ where $$ \alpha ...
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2answers
43 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
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2answers
41 views

Having trouble verifying a solution for a differential equation

Verify that $x=(t+1)e^{2t}$ is a solution for $$x = 2x+e^{2t},\ \ x(0)=1$$ My approach so far is. $$x' =2x+e^{2t}$$ $$dx/dt = 2x+e^{2t}$$ $$\int(dx-2x) =\int e^{2t}dt + C$$ $$-x^2 = e^{2t}/2 + C$$ ...
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0answers
22 views

Integrate multi-variable autonomous ordinary differential equations using Runge Kutta 4

I have a first-order ordinary differential equation (ODE) of the form: $$ \mathbf{\dot{y} = A\cdot y+B\cdot u} $$ where $\mathbf{y}$, the state variable, is a $7\times 1$ vector; $\mathbf{u}$, the ...
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0answers
17 views

Is this end-point map surjective

Consider the differential equation: $\frac{d U_s}{dt} = (a + w(s)b)U_s$ where $w$ is some unknown, smooth, real and bounded function on the interval $[0,T]$ and $a,b \in \mathfrak{su}(n)$. Let ...
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2answers
35 views

ODE using Laplace transform

[ I got my Y(t) to be : $$12 \, e^{-4} \, e^{-2s} \, [\frac{1}{12(s+2)} + \frac{1}{4(s-2)} - \frac{1}{3(s-1)}] + \frac{1}{(s-2)} - \frac{1}{(s-1)}.$$ so i assume I need to use t shifting for the ...
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0answers
12 views

Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
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0answers
18 views

Tough NL Diff Eq.

I'm trying to explore $$ \left( y'' + (1/x) \, y' \right)(1-y) \, – \, (1/x)\left(y'\right)^4 = 0 $$ with the initial conditions $y(0) = 0$ and $y'(0) = 1$. By substitution I can show that an ...
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0answers
25 views

In initial value second order DE problem, should the 2 conditions be at the same $x_0$?

Let's say that I have DE of $y''+p(x)y'+q(x)y=0$. To pick a particular solution, should the two conditions be [$y(x_0)=k_1$ and $y'(x_0)=k_2$]? or can be any other combinations of: [$y(x_0)=k_1$ and ...
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0answers
40 views

Issue in first order differential equation

I've tried many times to reach the solution of a first order differential equation (of the last equation) but unfortunately I couldn't. Could you please help me to know how did he get this solution. ...
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0answers
48 views

Second-order nonlinear differential equation

I am trying to solve the following differential equation: $ \ddot{x}(t) + a\ |\dot{x}(t)|^n\ sign(\dot{x}(t)) + b\ x(t) = c\ sin(\omega\ t) $ where $n$, $a$, $b$, $c$, $\omega$ are constants, ...
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0answers
16 views

Use the lemma in this section to show that if T is an invertible linear transformation

Use the lemma in this section to show that if T is an invertible linear transformation then ||T||> 0 and ||T^-1|| is greater than or equal to 1/||T||. Lemma: For S, T in L(ℝ) and x in ℝ 1.|T(x)|is ...
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0answers
24 views

Problem with initial values ODE

EQ = $y'+2xy=x$ Initial Value=$y(0)=-2$ $y'+2xy=x$ = $y'+y = \frac{1}{2}$ The solution of the Diff Equation $\frac{1}{e^x}$ $\int{\frac{1}{2}}e^xdx$ = $\frac{1}{2}+c$ I wonder how to check if this ...
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2answers
43 views

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

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
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1answer
27 views

Difficult Differential Equation ($2^{nd}$ order ODE) [on hold]

Solve $y''+\frac{x^2}{1-x^2}y=0$ over the domain $-1<x<1$.
3
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1answer
23 views

Find piecewise constant function u for $X'(t)=AX(t) + Bu(t)$ and $X(t)=\begin{pmatrix}10 \\0 \end{pmatrix}$ for some T

Consider the system $$x''(t)=u(t)$$ such that $x(0)=100, \; x'(0)=50$. Find a function $u$ piecewise constant such that $x(T)=0, \; x'(T)=10$ for a time $T$ Using the control theory language, it is ...
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2answers
59 views

Solving Simple Partial Differential Equation

I can't solve this partial differential equation. $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ The short answer in the book which i read from it , ...
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0answers
15 views

Conformal mapping for constant Gauss Curvature

The Sine-Gordon equation describes varying angles, conserving differential lengths in a mapping with constant Gauss curvature by means of an ODE. In which conformal mapping (conserving angles), can ...
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1answer
32 views

matrix differential equation and its stability

I have a differential equation of a $n\times n$ real matrix $X$: $$\dot{X}=-AX$$ $A$ is also a $n\times n$ real matrix. Two questions: 1) What conditions should $A$ satisfy if we want that $X=0$ be ...
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1answer
31 views

Solution of $xu_x + yu_y = 0$

I have the first oder PDE $$xu_x + yu_y = 0 \; \text{on} \; \mathbb{R}^2$$ and I found the solution of that PDE is $$u(x,y) = f\left(\frac{y}{x}\right) = e^C = K$$ which is a constant solution. So, ...
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1answer
23 views

Problem with initial values (Differential equations)

So i'm trying to solve a trivial problem but sadly I'm not good with math and i need help. SO I solve this equation $y'+y=2$ the solution was $2$, and the initial value $y(0)=2$. How can I check ...
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2answers
36 views

Discrete time equivalent to ODE

I'm reading a paper in which it is noted that $$\frac{dv(t)}{dt} = f(t) - \varepsilon v(t)$$ has the discrete time equivalent $$v(t+1) = v(t)\exp(-\varepsilon) + \frac{f(t)}{\varepsilon}[1 - ...
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1answer
28 views

analytical solution of a nonlinear differential equation

can we find a closed form solution -- such as a series solution -- of the following equation $$\frac{dy_0}{dt}+b\left(\frac{20}{27}y_0(t)^2+\frac{10}{27}y_0(t)-\frac5{81} y_0(t)^3-\frac4{81}\right) ...
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1answer
34 views

differential equations solvable only by numerical methods [on hold]

What kind (a general formula would be nice) of differential equations do not have solutions expressible explicitly or implicitly or by an integral sign? In other words, what kind of differential ...
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1answer
34 views

Finding the differential equation, given a solution

I am unable to understand how to find the differential equation when a general solution has been given. Here are a few example solutions, which require their differential equations to be found: (a) ...
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1answer
15 views

Constant solutions and uniquenss of solutions theorem for IVPs

What role do constant solutions play in the existance and uniqueness theorem? For instance, consider the IVP $$\frac{dy}{dx} = x$$ $$ y(0) = 0 $$ Clearly, this IVP has a solution in the form of $y ...
0
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1answer
27 views

Exact differential equation problem

I was finding the solution of a differential equation. But I'm stuck on this part. I tried simple integration but answer is incorrect. I don't know how to solve this. $$ dz=(6x+3y)dx+(3x-4y)dy $$
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1answer
13 views

Singular points while differentiating a function with respect to another function

I have $z(x) = \frac{df(x)}{dx}$ where $f(x)$ if a function of x. I'd like to have the derivative of $z(x)$ in respect to $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...