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

learn more… | top users | synonyms (1)

0
votes
0answers
7 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 ...
1
vote
1answer
13 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 ...
3
votes
0answers
12 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 ...
0
votes
1answer
18 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 $$ ...
2
votes
1answer
28 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 ...
3
votes
2answers
39 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 ...
1
vote
2answers
39 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. $$x' =2x+e^{2t}$$ $$dx/dt = 2x+e^{2t}$$ $$\int(dx-2x) =\int e^{2t}dt + C$$ $$-x^2 = e^{2t}/2 + C$$ ...
0
votes
0answers
19 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 ...
0
votes
0answers
16 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 ...
1
vote
2answers
34 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 ...
0
votes
0answers
11 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 ...
2
votes
0answers
15 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 ...
0
votes
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 ...
0
votes
0answers
39 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. ...
-1
votes
0answers
42 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, ...
-1
votes
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 ...
-1
votes
0answers
23 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 ...
1
vote
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 ...
0
votes
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 $-1<x<1$.
2
votes
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 ...
1
vote
2answers
54 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 , ...
1
vote
0answers
14 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 ...
0
votes
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 ...
0
votes
1answer
29 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, ...
0
votes
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 ...
2
votes
2answers
34 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 - ...
0
votes
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) ...
0
votes
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 ...
0
votes
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) ...
0
votes
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
votes
1answer
26 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. $$ dz=(6x+3y)dx+(3x-4y)dy $$
1
vote
1answer
11 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 $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
1
vote
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 ...
4
votes
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 ...
0
votes
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 $1/y^2$ ...
1
vote
0answers
13 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 ...
0
votes
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}, ...
0
votes
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 ...
2
votes
1answer
41 views

The differential equation $\frac{dy}{dx} +y^2 + \frac{x}{1-x}y = \frac{1}{1-x}$

I am learning how to solve differential equations and making some progress. However, how can one solve this example? The task is to find the solution to the equation $$\frac{dy}{dx} +y^2 + ...
1
vote
4answers
77 views

The differential equation $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;$

I am learning differential equations and can do the basic examples. However, how can you solve the differential equation $$\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;?$$
0
votes
0answers
58 views

Integrating Factor. [duplicate]

$(axy^2 + by) dx + (bx^2y + ax) dy=0$ I have asked this question before too, but i wish to know the method for evaluating the integrating factor which is $\frac {1} {(a-b)(x^2y^2-xy)}.$ So far i ...
1
vote
1answer
17 views

how to write a function in terms of Heaviside step function

I'm reading Paul Online Notes. There's an example of writing a function in terms of Heaviside step function as follows: $$ f(t) = \begin{cases} -4 &\text{if } t < 6, \\ 25 &\text{if } 6 \le ...
3
votes
0answers
36 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)}{ ...
0
votes
2answers
35 views

General ODE question

Find the general solution $y(t)$ of the ordinary differential equation $$y''+\omega^2 y=\cos \omega t,$$ where $\omega>0$. I'm relatively new to ODEs and PDEs but can someone show me the ...
0
votes
0answers
21 views

Love's equation $f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$

Let us consider Love's equation: $$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$ Is $f(x)$ a two times differentiable function?
0
votes
1answer
17 views

Find $\varphi_{1}$ from $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$

I have $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$ where $q_{1}=0$ and $\dot q_{1}=v_{0}$ and I must find $\varphi_{1}$. I know that $\varphi_{1}$ must be zero but I must demonstrate it first. ...
1
vote
1answer
30 views

How to solve differential equation $3p^2e^y-px+1=0$ ,$p =\frac{dy}{dx}$

How to solve differential equation $$3p^2e^y-px+1=0$$ where $$p =\frac{dy}{dx}$$ I have tried to solve for p and for x, but i am not getting anywhere. Can someone help me with this Thanks
0
votes
1answer
45 views

Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem $y'=y^2$, $y(0) = 1$, exist for all real $x$. (b) Find a solution of the initial value problem in (a). On what interval does it ...
2
votes
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: ...
0
votes
0answers
22 views

Determine for what values of m the function is a solution

I was working through some differential equations and came across this problem. Determine for which values of $m$ the function $\phi(x)=e^{mx}$ is a solution to the given equation. A) ...