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

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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
38 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
27 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
19 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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0answers
10 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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1answer
15 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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0answers
13 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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13 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
47 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
12 views

Problem on vertical light elastic string

A mass of $4$ lbs suspended from a light elastic string of natural length $3$ feet extends it to a distance $2$ ft. One end of the string is fixed and a mass of $2$ lbs is attached to other. The ...
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1answer
27 views

Finding the Equations of Motion for the Leapfrog Integrator

I understand that the Leapfrog Integrator is used to find an integral for Newton's Laws of Motion and that the Equation of Motion are given by: $$\frac{dx}{dt} = v$$ and $$\frac{dv}{dt} = F(x) = ...
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2answers
47 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
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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1answer
12 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 $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
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0answers
11 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
14 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 ...
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1answer
19 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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0answers
18 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
47 views
+50

Showing a bound exists

I was able to derive the following differential equations I have to work with for a function $V$: $$ \begin{align*} dV(x_1,x_2,x_3,x_4) &= ...
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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
621 views

Spring Calculation - find mass

A spring with an -kg mass and a damping constant 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 newtons. If the spring is stretched 5 meters beyond its natural length ...
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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
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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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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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1answer
3k views

Solve a second order DEQ using Euler's method in MATLAB

I need to solve the equation below with Euler's method: $$y''+ \pi ye^{x/3}(2y' \sin(\pi x)+\pi y\cos (\pi x)) = \frac{y}{9}$$ for the initial conditions $y(0)=1$, $y'(0)=-1/3$ So I know I ...
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1answer
22 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 ...
2
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1answer
76 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
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1answer
58 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using Fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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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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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
17 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
19 views

*Solved* Terminology in DE, difference between Particular and Actual solution

Yesterday I started studying and preparing for a course in Differential Equations and today I came across something that confuses me; I watched a lecture on IVP and they used both Actual solution and ...
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37 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
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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
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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2answers
40 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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0answers
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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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3answers
194 views

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, ...
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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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2answers
67 views

How do I find the exact solution to the boundary value problem $y'' = 4y' + y + 2 − 8x − x^ {2}$ , $y(0) = 0$ and $ y(4) = 16$?

I am approaching this question by trying to guess the general solution to the boundary value problem. However I haven't come up with one. Can someone explain how to solve this question please?
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2answers
130 views

How to apply the Gronwall lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
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1answer
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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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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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2answers
130 views

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
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1answer
73 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
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1answer
30 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
27 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
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 ...