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

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4
votes
1answer
41 views

Is this differential equation separable?

$$x\frac{dy}{dx}-y^2 = \frac{dy}{dx}+5$$ I have found that this equation is differentiable as shown in the following. $$x\frac{dy}{dx}-\frac{dy}{dx} = y^2+5$$ $$dy(\frac{x}{dx}-\frac{1}{dx}) = ...
0
votes
1answer
28 views

Sketching phase portrait

$\dot{x}=-2x-2y$ $\dot{y}=-x-3y$ Equilibrium point is $(0,0)$. Eigenvalues are $\lambda_+=-1$ and $\lambda_-=-4$ which have corresponding eigenvectors $2\choose -1$ and $1 \choose 1$ respectively. ...
0
votes
0answers
20 views

How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
1
vote
1answer
26 views

How to guess a change of variable for an ODE?

My question is related about how to guess what change of variable to make in an ODE? For example, we have $$\varepsilon^2y(x)''=axy(x)$$ where $\varepsilon\ll1$, $a$ is constant and $x$ is the ...
0
votes
0answers
11 views

Solving System of Boundary Value problem

The boundary value problem: $$y'' + Q(t)y = f(t)$$ satisfying $$Ay(a) +By(b) = g$$ where A, B and Q are the matrices of order n. After calculation, we can get the form of solution will be $$y(x) = ...
1
vote
0answers
17 views

Solving IVP using Laplace Transform

Let $$g(t) =\begin{cases} t & \text{if $t \leq6π$} \\ 6\pi & \text{if $t>6\pi$} \end{cases} $$ Solve $y''+ 16y = g(t)$ where $y(0) = 9$ and $y'(0) = 4$ using Laplace transforms. I got ...
2
votes
0answers
19 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
5
votes
1answer
61 views

Solving $y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}$

There is the trivial $y=0$, but beyond that, could there be further solutions for $y$ in terms of $x$ such that $$y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}\mbox{ pointwise}$$ ? I posed this problem ...
0
votes
0answers
26 views

Inverse laplace transform of $1/(s^2 +1)^{1/2}$

I tried this problem using the basic properties but I am unable to get further than reducing this one to $(s^2+1)^{1/2}$.
2
votes
1answer
19 views

Lipschitz condition in non-autonomous case

I need to show the RHS of $\frac{dy}{dt}=-y^2+y+2yt^2+2t-t^2-t^4=f(t,y)$ is locally lipschitz. Am I able to use continuously differentiable implies locally lipschitz here with non-autonomous $f$? ...
0
votes
1answer
24 views

Series solution of a differential equation with a function as a coefficient?

I'm trying to solve a differential equation using series: the equation is $$y''+(\sin(x))y=0$$ I know you have to use the power series form of $\sin(x)$ and multiply it with the power series form of ...
1
vote
1answer
36 views

Black-Scholes Problem

Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 \hspace{3mm}\textrm{and}\hspace{3mm} ...
0
votes
0answers
31 views

Meaning of maximal interval of existence [on hold]

Does this mean the entire domain for which the solution of the IVP is continuous on before it blows up?
0
votes
1answer
34 views

Fundamental solution of a matrix

I want to find the fundamental solution of $x''+\dfrac{x'}{t}-\dfrac{4x}{t^2}=0$. I have converted to its equivalent linear homogeneous system including variable $y(t)$: $$\begin{bmatrix} x'(t)\\ ...
-1
votes
1answer
43 views

Solve the differential equation by variation of parameters. $y'' + y = \sin x$ [on hold]

Solve the differential equation by variation of parameters. $y'' + y = \sin x$ What I have so far: $r^2+1=0$ $r^2=\sqrt{1}$ $y(p)= C_1e^{-x}+C_2e^x$ than i'm not sure what to do.
0
votes
1answer
24 views

Separation of variables and basis of solutions?

If a PDE can be solved by separation of variables. Then the superposition of the solutions found via this method can form all other solutions to the PDE. Is this statement correct? If it is ...
1
vote
1answer
26 views

Question on Fredholm alternative in an ODE

I am playing with this example and try to understand how to use Fredholm alternative theorem to determine what values of $\beta$ that yield existence of a solution to this problem. And if solutions ...
3
votes
2answers
64 views

Riccati Equation for falling particle.

I'am trying to solve the differential equation for a falling particle of mass 1 with air resistance proportional to $v^2$ (v is velocity): $$v'=g-v^2$$ This is a Riccati-Equation with stationary ...
0
votes
1answer
33 views

Solutions cannot cross

I understand for for the initial value problem: $\frac{dx}{dt}=f(x) \quad$ $x_0=x(0)$ If I have two solutions $x_1(t),x_2(t)$ defined on the same interval with the same initial condition satisfying ...
3
votes
1answer
25 views

Lipschitz condition not satisfied

To show there is no contradiction to existence and uniqueness $\displaystyle\frac{|f(x,u)-f(x,v)|}{|u-v|}= \displaystyle\frac{|x||u^{1/2}-v^{1/2}|}{|u-v|}=\frac{|x|}{u^{1/2}+v^{1/2}}$ I understand ...
0
votes
4answers
40 views

Eliminating $t$ in the solution of a Differential Equation

My task is to show that the trajectories of the system: $\frac{dx}{dt}=y$, $\frac{dy}{dt}=x$ are hyperbolas given by $H(x,y)=y^2-x^2=c.$ Solving the above system I got: $x=c_1e^t+c_2e^{-t}$, ...
1
vote
3answers
44 views

Converting nonlinear second order ODE to first order

The second order ODE is: $$ y'' + ky' + (mg/c)\sin(y) = 0 $$ $k$, $m$, $g$ and $c$ and constants. What is the first order ODE and how do I do it?
0
votes
1answer
23 views

Determining properties of Solutions of an autonomous ODE

Which of the following functions is a solution of a differential equation of the form $$y'=f(y)$$ where $f$ is continuously differentable for all real $y$ (a) $y=sin(t)$ (b) $y=cos(t)$ (c) ...
0
votes
1answer
16 views

Limits of coefficeints in a system of O.D.E.s.

Say that we have a system of O.D.E.s that depend on some real parameter $c$ \begin{equation*} \dot{x}_i = f_i(c,x_1,...,x_n) \ ,\ \ \ i=1,2,...,n \ . \end{equation*} I've not really seen what it ...
0
votes
1answer
27 views

How do we plot nonlinear differential equations

If this is not nonlinear I apologize, I'm still learning differential equations. I am attempting to make a stream plot of a predator-prey model of eccentric closed curves by using the following ...
-1
votes
1answer
19 views

Ordinary Differential Equations General Form [on hold]

What are all solutions of $f''(x)=\lambda f(x)$ on $0\le x \le L$.
1
vote
1answer
53 views

$\psi ( \frac {\pi}{2}, \frac {\pi}{6})$ and calculating problems?

I ran into a problem, $u=\psi (x,t)$ be a solution of partial deferential equation with following condition on boundary, how we reach the value of $\psi ( \frac {\pi}{2}, \frac {\pi}{6})$? ...
3
votes
0answers
71 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in Polar coordinate $ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial u} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2}$ with ...
2
votes
2answers
34 views

How to guess the graph of a differential equation?

Can anyone provide some techniques for solving this kind of problem? Many thanks.
0
votes
0answers
23 views

An example of a linear system that has a straight-line solution $[x(t), y(t)]$ such that $x(0) = -1$ and $y(t) = 2x(t)$ for all $t$

Give an example of a linear system that has a straight-line solution $[x(t), y(t)]$ such that $x(0) = -1$ and $y(t) = 2x(t)$ for all $t$. What I Know: Equilibrium solutions are constant ...
1
vote
1answer
26 views

How to show that if $f$ is a solution for $y" + y = 0$ and the graph of $f$ passes contains 2 arbitrary points then f is unique

Suppose that $f$ satisfies $$y'' + y = 0 $$ And the graph of $f$ contains the points $(a_{1},b_{1})$ and $(a_{2},b_{2})$, with $$a_{1}-a_{2} \neq n \pi, n \in \mathbb{Z}$$ Show that $f$ is the ...
1
vote
0answers
43 views

Study the stability of the following ODE: $u'=u(1-u)-\alpha$

Given that $\alpha$ is real, I'm being asked to give a basic analysis for this nonlinear ODE. However the problem is that I'm having trouble understanding how to pick the conditions we need to study ...
2
votes
1answer
18 views

Question regarding $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$

The book asks for: $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$ So I can see: $\frac{s}{s^2+4s+4+1} = \frac{s}{(s+2)^2+1}$ From the properties of the inverse Laplace transform: ...
0
votes
2answers
28 views

Finding solutions for the differential equation: $x(y^2-7)-(x^2-6)yy'=0.$

Given the differential equation: $x(y^2-7)-(x^2-6)yy'=0.$ How do I find a solution for the equation that satisfies $y(1)=-2$ (if it exists) or $y(\sqrt{-6})=-2$. I know that I need to solve this ...
1
vote
2answers
22 views

Globally Lipschitz implies solutions exist for all time

I understand when you have a locally lipschitz functiion $f$ for the IVP you are guaranteed the existence and uniqueness of a solution. However this solution may not exist for all time. Why is it that ...
-1
votes
2answers
59 views

How can I solve the following differential equation without use power series [on hold]

Let we have the following differential equation $$y''-xy'=e^{-x}$$ how can I solve this differential equation without use power series
0
votes
0answers
17 views

One-dimensional Boundary Value Problem with complex coefficients and mixed Dirichlet-Robin boundary conditions

I am trying to solve the following pair of Boundary Value Problems for the unknowns $\phi^+(x)$ and $\phi^-(x)$, on the line $0 \leq x \leq L$: $\phi_{xx}^{\pm} \mp \frac{il}{k}\phi^{\pm} = 0$ ...
4
votes
2answers
119 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
-1
votes
2answers
25 views

Uniqueness of a solution to an IVP over a large domain

I get that both $x_1(t)$ and $x_2(t)$ are continuous on $I$ as they are differentiable on $I$ as they are solutions to the differential equation. However I do not understand why their continuity ...
0
votes
0answers
23 views

Connection between these two ODE

We have the following discrete time difference-equations: $$ F(x_t) = \Delta G(x_t) + (1-a \Delta)F(x_{t+\Delta})\\ x_{t+\Delta} = \underbrace{\Delta b(1-x_t) + \Delta x_t f(F(x_t))}_{\Delta H(x_t, ...
0
votes
1answer
50 views

How is this concluded in the following differential equation?

Question: Solve the differential equation $$t^2x''-2tx'+(t+2)x=t^6$$ Answer:$$x(t)=u(t)v(t); x'=u'v+uv' ; x''=u''v+2u'v'+uv'';$$ $$t^2(u''v+2u'v'+uv'')-2t(u'v+uv')+(t^2+2)uv= t^6 $$ then: ...
-1
votes
0answers
31 views

Initial value problem with unique solution and rear wheel of a bike problem

Consider $I\subseteq\mathbb{R}$ an arbitrary interval (it can be of the following types: $[a,b], [a,b), (a,b], [a,+\infty),(a,+\infty),(-\infty,b), (-\infty, b], \mathbb{R}$, where $a,b\in\mathbb{R}, ...
2
votes
4answers
220 views

Differential equation with integration factor

I tried to solve this differential equation: $$ydx+(2xy-e^{-2y})dy=0$$ I found $e^{2y}$ as integration factor but when affect this on equation I don't get $M_{y}$=$N_{x}$ (they are not exact)... ...
1
vote
0answers
20 views

Picard Iteration

For part b) I understand how to get to $z_1= \begin{pmatrix} t \\ 1 \\ \end{pmatrix} \quad$and because $z= \begin{pmatrix} x \\ p \\ ...
0
votes
2answers
25 views

A differential maximization problem

OK, I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a 'maximization problem' over functions? ...
-1
votes
0answers
31 views

Why use the Runge-Kutta method to solve differential equations? [on hold]

Why does one use the Runge-Kutta method to solve differential equations, calculating previous and future positions? Why not use, for example, normal equations, derivatives, or something else?
2
votes
1answer
40 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
0
votes
1answer
22 views

How to use the crank-nicolson method

I'm going over my study questions for an exam I have tomorrow in Applied Numerical Methods and I know everything except for one thing. There's a sample question about using the Crank-Nicolson method, ...
1
vote
1answer
24 views

Where do I begin to solve this ODE?

I've a problem: $ye^{xy}+4y^3+(xe^{xy}+12xy^2-2y)y'=0 $ I can't begin to solve it because I don't know where to start. Help me, please. Thanks!
1
vote
2answers
21 views

What is my error in applying this Laplace Transform?

So, our book has the seemingly innocuous problem: $y''-y'-6y=0$. I was able to solve by hand, and come up with $${\scr L}(y)=\frac{s-2}{s^{2}-s+6}$$.That completed, I factored the bottom to ...