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

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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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1answer
11 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
21 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
28 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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14 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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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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15 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
40 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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17 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
16 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
49 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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12 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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0answers
19 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
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
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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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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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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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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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
46 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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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
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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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 $-1<x<1$.
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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 ...
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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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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
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
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 ...
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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
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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1answer
33 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
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1answer
47 views

Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
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0answers
33 views

Why don't we check the exactness of differential equation with Inspection cases?

When solving the differential equations which are reducible to exact differential equations using Inspection cases for example: Solve: $2xy^2 + ye^xdx = e^xdy$ The integrating factor would $1/y^2$ ...
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0answers
14 views

Using the method of isoclines with logistic equation to create direction field

I am a little unsure on how to use the method of isoclines to model $\frac{dp}{dt} = 3p-2p^2$. As far as I know I need to set $3p-2p^2 = c$ where $c$ is the slope of the field on that line. When I set ...
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1answer
25 views

Lowering the order of a linear differential equation

Let $$L(x) \equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ and let the following solutions be given: $x_1,x_2,...,x_m(m<n)$- linear independent solutions. Let's find: $x_{m+1}, ...
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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 ...