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

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3
votes
0answers
25 views

the solution of $\lambda u''(x) = u(x)$ is $u(x)=\{sin(n\pi x)\}_{n=1}^\infty$

in my text it says: the solution of $\lambda u''(x) = u(x)$ is $u(x)=\{sin(n\pi x)\}_{n=1}^\infty$with boundary condition u(0)=u(1)=0 how do I know that this set contains all solutions? What if their ...
4
votes
3answers
63 views

How do I solve the following differential equation

$$\frac{d^2y}{dx^2}=x^2y$$ Solving it by writing out a characteristic equation is not helping me find the solution to the above equation. Any help would be appreciated thanks.
-1
votes
1answer
33 views

Working out rent?

I have a question that I'm trying to work out and the explanation behind the answer would be great: A company has paid rent of £16,000 during the year including all this years rent and a payment for ...
1
vote
1answer
22 views

Looking for tip/procedure of series solutions to ODE

I have been having a few questions about series solutions to ODE and I found an example that can illustrate my question. It is just a simple example, say we consider the ODE $$ y''-xy'-y=0$$ around ...
0
votes
0answers
23 views

Calculating the magnetic vector potential

I've calculated A as proportional to r^2 for ra but I really don't think is correct. If someone could take me through the calculation I would really appreciate it.
-1
votes
1answer
23 views

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? [on hold]

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? If so, how are they essentially different from each other?
-2
votes
1answer
23 views

Solve the initial-value problem by Laplace [on hold]

I must solve the initial-value problem $$ \begin{cases} 2y'' + 5y'- 3y = 0\\ y'(0)= 31\\ y(0)= -1 \end{cases} $$ How am I supposed to do this?
2
votes
3answers
42 views

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$ my work $f'(x)=\tan^{-1}(x\ln(x)) *1 + x$ ---> stack here I know $\tan^{-1}(x)'= \frac{1}{1+x^2}$ so $\tan^{-1}(x\ln(x)) = ???$ I need help to solve ...
0
votes
0answers
8 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
0
votes
0answers
8 views

Differentialequation with Eulers method

I have a problem with a differential equation that can be used Euler method in a digital manner. I use a program that is designed to excel. The entire task looks like this: Differential equations y ...
2
votes
0answers
27 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
0
votes
2answers
24 views

How to plot a phase portrait for system of differential equations in mathematica or R?

Please, help me. I'd like the phase portrait for this system: If anyone can make this portrait and post a print screen here, I would thank you very much.
1
vote
0answers
19 views

System of ODEs: Boundary Value Problem in Matlab

I want to finde a numerical solution for the following 3d system of time dependent ODEs \begin{align} \dot{y}_1 &= y_3 + \frac{y_1}{20} + \frac{1}{y_3+y_1} - \frac{1}{1-y_2} -\frac{1}{y_3} - ...
2
votes
0answers
17 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
-3
votes
2answers
43 views

Differential equations/ 4 [on hold]

How to solve this differential equation: $$ \frac{ydy+zdz}{\sqrt{y^2+z^2}}+\frac{ydz-zdy}{y^2}=0$$ I gave similar but then nothing happens maybe this is exact differential equations?
-4
votes
1answer
58 views

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ [on hold]

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ Consider a 1st order differential equation $dy/dt + 409y = f(t)$ a)Solve and write the general ...
2
votes
1answer
45 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
1
vote
1answer
23 views

Differential Equations: Stable, Semi-Stable, and Unstable

I am trying to identify the stable, unstable, and semistable critical points for the following differential equation: $\dfrac{dy}{dt} = 4y^2 (4 - y^2)$. If I understand the definition of stable and ...
3
votes
2answers
57 views

How to determine generalized eigenvectors of $\begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 &1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$

I want to calculate the general solution of this DE-system: $$ \frac{d \vec x}{d t}= A \vec x,\text{ with }A = \begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 ...
0
votes
1answer
22 views

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? [on hold]

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? Any help or just tips will be appreciated, thanks.
1
vote
1answer
41 views

how to show that all solutions tend to zero?

Here is our nonlinear first order ode: \begin{equation*} y'(t) +2y(t)+y^3(t)=e^{-t} . \end{equation*} We want to show that all solutions tend to zero as $t$ goes to infinity. Attempt: Multiply both ...
-1
votes
2answers
29 views

System differential equations 0 [on hold]

System of nonlinear differential equations $$y'= -\frac{4y}{x+4}+\frac{y^2x}{4t}, $$ $$ x'= \frac{x^2}{t^2}-\frac{9x}{t}+24 $$ help if I understood correctly you need to express $x$, but I can't
1
vote
2answers
53 views

Sane solution for an ODE with physical interpretation

I have an object which is being subjected to a continual force that is a quadratic function of the object's velocity, ie, $F=f_0+f_1 v + f_2 v^2$ for arbitrary but given constants $f_0$, $f_1$, and ...
0
votes
1answer
18 views

Question about phase shift on multiple-scale analysis

Consider the following ODE $$y''(t) + y(t) + \epsilon y^2(t) y'(t) = 0$$ for $t>0$ with boundary condition $y(0)=1$ and $y'(0)=0$ I have found the leading order asymptotic expansion, that is ...
-1
votes
0answers
29 views

first-order nonlinear ordinary differential equation0 [on hold]

How to solve this differential equation: $$(x^{2}+\ln(y))\cos(2x)+\sin(2x)(xdx+\frac{dy}{2y})=0 $$ I tried to rearrange the equation to the form $\frac{dy}{dx}$ but I couldn't.
0
votes
1answer
14 views

Learning spectral methods in numerical analysis

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
0
votes
0answers
26 views

limit problem-equation

H, I have this problem $$c^2 U''(x)=F(x),\quad U(0)=A,\quad U(\ell)=B$$ $F$ is done, and $0 < x < \ell$ I read that we must found that $$U(x) = A + (B-A)\frac{x}{\ell} + \dfrac{x}{\ell} ...
0
votes
3answers
37 views

Particular solution of y'' -3y' + 2y = e^t

I'm trying to find a particular solution of $$y'' -3y' + 2y = e^t$$ My fundamental set is: $$y_1 = e^{2t}\\y_2 = e^t$$ So I chose $y_p = A t e^t$, which gives me:$$y_p' = Ae^t + Ate^t\\y_p'' = 2Ae^t ...
1
vote
1answer
24 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
0
votes
0answers
11 views

Do you know how get differential equations of HSIR model of propagation malware? [on hold]

I have differential equations but I don't know how get it? thank you for help me.
0
votes
3answers
46 views

Solve the following second-order differential equation: $\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t)$

I am trying to solve the following second-order differential equation: $$\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t). (*)$$ I know that if the equation had instead been: $$\ddot{x} + \dot{x} = ...
0
votes
1answer
18 views

How to find solution to $y'=y_1(x)g(x)+y_2(x)f(x)$?

How to find solution to $$y'=y_1(x)g(x)+y_2(x)f(x)$$ Asuume that function $y=y_1(x)$ is one of the solutions of differential equation $y'=f(x)$ as well as $y=y_2(x)$ of $y'=g(x)$. You need to ...
0
votes
0answers
39 views

Confusion regarding dF/dx=0, F=constant

I thought i found a theorem "Given a curve in the (y,x) plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of F along this curve satisfies relation $g = ...
0
votes
1answer
20 views

Bessel Functions of Half-Integer Order

I recently came across the general form of Bessel Functions of half-integer order given by: $$ ...
2
votes
2answers
26 views

Need help with linear ODE, indicial and recurrence.

I am having trouble understanding something and I want to post what I have done so hopefully someone can catch where I have made a mistake. The question asks; determine the indicial equation, ...
-5
votes
0answers
17 views

Let v take any arbitrary value define y extraneously [on hold]

Verify: f'(x) O f'(y) e^x ln 2x = bc Tan (2x)^v(4C2). Quite easy but lengthy. here, Let v take any arbitrary value define y extraneously
2
votes
1answer
35 views

Equilibrium solutions for $y'=t^{3}y$

I'm having trouble understanding the following. To solve the differential equation $$y'=t^{3}y$$ I go about it in the following way: \begin{align*} y'&=t^{3}y\\ \frac{y'}{y}&=t^{3}\\ ...
2
votes
1answer
55 views

Confusion about ODE

so I am in a class for ODE and for me is is moving a bit quick. I am one year behind most of the class but thats note anything rare. But I am feeling very stumped on something now. Because, usually I ...
2
votes
0answers
37 views

Continuation of differential equation

Suppose I have a differential equation $$\dot{x} = f(x)$$ which has global solution for any initial value $x(0) \in \mathcal{S}$. Is there some theorem defining conditions under which this equation ...
1
vote
0answers
19 views

Exact Similarity Solutions of System of Nonlinear Partial Differential Equations [on hold]

I have been reading Self-Similarity and Beyond, by P. L. Sachdev. However, I am stuck on page 70, chapter 3, section 2. I have screen shotted the part which I am having a problem with I wonder if ...
0
votes
3answers
33 views

Determine the form of solution to differential equation, for particular starting value

I am working on a differential equations problem. I must first find the general solution to: $$y' = y(y-1),$$ where $x$ is the dependent variable. I have managed to solve this, to get the answer: ...
3
votes
1answer
55 views
+50

Behaviour of solutions to ODE near singular points

I am having trouble understand how to classify what happens to solutions of ODE near singular points. For example, I have a question that is about the ODE; $$(x^2-36)y''+(6-x)y'+(x^2+12x+36)y=0$$ ...
3
votes
1answer
38 views

How do I go about solving this differential equation?

$$t^2x''-(6t^4+2t)x'+9t^6x=0$$ I was taught to write as the following $x= t^n+a_1t^{n-1}... \\ x'=nt^{n-1}+a_1(n-1)t^{n-2}...\\ x''=n(n-1)t^{n-2}+a_1(n-1)(n-2)t^{n-3}...$ And then plug those into ...
0
votes
1answer
33 views

Laplace diffrential equation

$$\frac{dx}{dt}=2x +3y$$ $$\frac{dy}{dt}=3x +2y$$ Find general solution. I know there is a solution through eigenvalues. But I want to solve it with Laplace transformation. I almost get the right ...
1
vote
1answer
17 views

Question about Frobenius Method

I am having some confusion and looking for some help/suggestions about the following. Consider the ODE; with regular singular point $x_0=0$ $$2x(x-1)y''+3(x-1)y'-y=0$$ And I am supposed to find the ...
0
votes
2answers
39 views

Solving a higer order differential equation

Let $n=1,2,3\dots$ Discuss how the observations $D^n(x^{n-1})=0$ and $D^n(x^n)=n!$ can be used to find the general solutions of the given differential equations. $y''=0$ $y'''=0$ $y(4)=0$ $y''=2$ ...
2
votes
2answers
36 views

Solve of the differential equation $y'=-\frac{x}{y}+\frac{y}{x}+1$

I've tried to solve this equation, and in the course of solving any problems. Please help me understand. $$y'=-\frac{x}{y}+\frac{y}{x}+1$$ Results in a normal form. ...
2
votes
2answers
30 views

Solve of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$

I have some problem. There is an equation: $$\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$$ Open brackets. $$3y^2dy+x^2dy+2xdx y+xdy +ydx +2ydy+dy=0$$ But what to do, tell me, please? I saw this a ...
-1
votes
2answers
46 views

How can I solve $y$ in differential equation? [on hold]

$xy'(x)=y(x)(x+1)$ where $y(1)=2e$ I've no idea whatsoever to begin and get an answer! Hints are welcome!
1
vote
1answer
26 views

Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation?

Is it true that the number of arbitrary constants in the solution (if solutions exist) always equal to the order of an ordinary differential equation? If yes, how to "prove" such a statement, if it ...