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

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Modelling interest with differential equations (Interpretation)

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows: Assume you have a bank account that grows at an annual ...
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0answers
10 views

General examples of Sturm-Liouville operators

The topic: My question pertains to examples of Sturm-Liouville operators in the context of a technical research paper on functional determinants of differential operators : ...
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0answers
6 views

Determine the error constant for $y_{n+2}-4\theta y_{n+1}-(1-4\theta)y_n=h\left[(1-\theta)y_{n+2}'+(1-3\theta)y_n'\right]$

I have the following problem but I cannot solve part B in the way suggested by my professor in this past exam paper. I can solve it in a different way, but not in the specific way he's suggesting. ...
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0answers
10 views

Criteria when bigger number of functions can be obtained from smaller number

It is known that $$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial ...
3
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1answer
27 views

Solve the following differential equation $ u_{xx}-m^2u=\delta(x-x_0)$

Find the solution of following equation $$ u_{xx}-m^2u=\delta(x-x_0),$$ $u(0)=0=u(L),\ x\in\mathbb R^2$ Actually, I don't know how to solve. Is there someone to help?
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1answer
48 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
2
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0answers
37 views

Is there a numerical solution for a system of three 1st order nonlinear ODE?

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$ $$x' = y $$ $$y'=\sin(x)+z$$ $$z'=y-z$$ I have the following initial conditions; $$x(0) = 0$$ ...
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1answer
24 views

Continuously differentiable functions are weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial ...
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0answers
28 views

how to solve an affine differential equation

Is there a general way to solve $y'=Ay+b$, with $y, b \in \mathbb{R}^n$, $A$ a matrix, and where $A$ and $b$ are constant? I'm tempted to make the substitution $z = y+A^{-1}b$, and then use the matrix ...
3
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0answers
24 views

Poincaré-Bendixson theorem, periodic solutions/periodic orbits

According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of: 1) $\omega(\phi)$ contains an equilibrium, or 2) either $\phi(t)$ ...
3
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0answers
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Exam question: Are zero points justified for this answer?

I just recently had an exam and had to answer the following question: Find the solution to the initial value problem $$x'(t)=\frac{1}{x(t)}; \space x(0)=1$$ and specify the maximum interval off ...
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0answers
24 views

How fast is the water level falling when the water level is 12 meters high?

Water is draining from a conical tank (with vertex down) at the rate of $2m^2/s$. The tank is 16 meters high and its top radius is 4 meters. How fast is the water level falling when the water level is ...
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0answers
16 views

Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
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Qualitative Ordinary differential equations [on hold]

Reduce the following systems of equation to a systems of first order ODE’s: 〖( d^2 y)/〖dt〗^2 〗^+3 dz/dt+2y=0 〖( d^2 z)/〖dt〗^2 〗^+3 dy/dt+2z=0
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3answers
31 views

Separating variables by substitution in a homogenous ODE

I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of ...
3
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1answer
36 views

Solving a SDE / Finding expectation Value

I am working on a physics problem, and have come across the following stochastic differential equation: $dX(t) = \left( \frac{8}{3} X(t) - 3 X(t)^3\right)dt + dW$. I have tried all the methods to ...
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1answer
12 views

What is a critical point in a system of equations?

I have an assignment question based around a system of nonlinear differential equations, $$ x' = f(x, y) \\ y ' = g(x, y) $$ The first part of the question is to locate and classify all the ...
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1answer
24 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
3
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1answer
42 views

Runge Kutta stability

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
1
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1answer
15 views

Flow of time-depended vector field

Suppose $X_t$ is a time-depended vector field with flow $\phi_t$, so, $\frac{d}{dt} \phi_t = X_t(\phi_t)$. Is it true that $d \phi_t(X_t(x)) = X_t(\phi_t(x))?$ This is true when $X_t$ does not ...
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1answer
34 views

Help with Euler Equations

This is from my textbook. Can someone give me a better explanation of what to do here? What does part (a) mean, i.e., how am I supposed to write $x = ln(t)$ in terms of $\frac{dy}{dx}$ and ...
3
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1answer
38 views

When can you take the limit of a parameter before solving the differential equation?

Short example: consider the differential equation \begin{align*} f'(x)=\frac{k^2}{k^2+k+1}xf(x) \end{align*} where $k$ is a parameter. Wolfram Alpha tells me that the solution to this equation is ...
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3answers
35 views

Forming differential equation

I'm trying to get from: $$e^{\lambda t} (\frac{dN}{dt} + \lambda N) = re^{\lambda t} $$ To: $$ \frac {d}{dt}(Ne^{\lambda t}) = re^{\lambda t} $$ However I'm not sure what procedure to use to go ...
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2answers
26 views

question on second -> first order systems [duplicate]

I have heard that it is possible to write second order IVP as first order system. What are some strategies to writing $y''=xy^2$, $y(0)=1$, $y'(0)=2$ as a first order system $y'=f(y)$, $y(a)=y_0$? ...
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1answer
62 views

Units of ODE solution don't match

I have to solve the differential equation: $v\,'=g-cv$. Sorry in advance for lack of latex. I will learn it soon, please let me make a question using the common programming notation for my ...
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0answers
26 views

Separation of variables, Homogeneous or Exact Differential equations?

So I've just encountered these three, during exams of course they don't tell you which one is to use, if you need to use separation or homogeneous or exact. I was just wondering is there like a ...
1
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4answers
92 views

Solve the equation $(m^2-m-2)x=m^2+4m+3$

Here's how I solve it I think that m is the variable (am I right?). Then $$m^2x-mx-2x-m^2-4m-3=0$$ $$m^2(x-1)-m(x+4)-(2x+3)=0$$ $$D=x^2+8x+16+4(x-1)(2x+3)$$ $$=x^2+8x+16+4(2x^2-2x+3x-3)$$ ...
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0answers
16 views

The number of characteristic curves of a pde

When a partial differential equation is elliptic, $B^2-4AC\lt 0$ and eigenvalues are complex. does there exist any characteristic curves?
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Simple related rates derivative question

Rafael is walking away from a $12$-ft-tall lantern at a constant speed. If the tip of Rafael's shadow is moving twice as fast as he walks, how tall is Rafael? I'm confused on the step where $dL/dt = ...
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1answer
18 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
6
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1answer
63 views

Flow of sum of non-commuting vector fields

Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of ...
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3answers
56 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
1
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1answer
21 views

Norm bound on exponential matrix with eigenvalue negative real part, proof

If $A$ is $n \times n$ with negative real parts of all eigenvalues, then there exists positive $K,\alpha$ such that $$\|e^{At}\| \leq Ke^{-\alpha t}$$ Furthermore, if an eigenvalue has negative part ...
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2answers
41 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
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1answer
69 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
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2answers
30 views

Solving $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$ for $x(t)$

I`ve been trying to solve the differential equation $x(t)'=x(t)(a-x(t)), x(0)=b, t\in [0, \infty]$. Using the technique of seperation of variables, I get $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$. Now ...
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2answers
33 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
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1answer
26 views

Ordinary differential equation . [on hold]

The roots of the auxiliary equation for a homogeneous linear differential equation with real constant coefficients that has $ y= 4 + 2x^2 - e ^{-3x}$ as a particular solution are : 1) $ m= 0 , 0 , ...
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0answers
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Proving this is the unique solution to this simple system of diff equations.

So the set of equations are these $\frac{d \omega_x}{dt}+\Omega \omega_y =0$ $\frac{d \omega_y}{dt} - \Omega \omega_x =0$ You can easily differentiate again, get two second order linear diff ...
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0answers
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how to rearrange matrix equation to have unknown in vector form

I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times ...
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1answer
19 views

Question about assumptions for Picard-Lindelof Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
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1answer
77 views

How to solve this DE?

Consider the ordinary differential equation $$y''=xyy'$$ I'm pretty stumped, so any tips on how to proceed? It seems fairly simple but I'm drawing a blank.
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3answers
62 views

Initial Value Problem $dy/dx = (y+1)^{1/3}$

Consider the differential equation $$\frac{dy}{dx} = (y+1)^{1/3}$$ (a) State the region of the $xy$-plane in which the conditions of the existence and uniqueness theorem are satisfied (using any ...
2
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2answers
29 views

Coupled second-order differential equations

I am trying to solve the following system of coupled ODEs: \begin{align} -x^2 f'' - 3xf' + (1-2a)f - (a+1)x^2g'' + (2-4a)xg' + (4a-2)g &= 0,\\ (a-1)x^2 f'' + (4a+2)xf' + (12-6a)f + 12xg' + ...
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3answers
222 views

Solution of Differential equation

Question: Find solution of differential equation $$ 3e^{4x} \frac{dy}{dx} = -16\frac{x}{y^2} $$ which satisfies the initial condition y(0)=1 Solution: I know that I have to bring it in the general ...
3
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1answer
32 views

homogeneous first order differential equation

is there a method to solve $$\dfrac{dy}{dx} = f(x,y)$$, where $f(x,y)$ is a homogeneous function. I found some examples like $f(x,y)=(x+y)^2$ where it can be solved after converting it to Ricatti's ...
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1answer
37 views

graphing the solution of $y'=x^2-3$

I have a Ordinary Differential Equation(ODE) and I got the solution as ​ Now I want to draw graph? How can I do that? I think: ...
5
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3answers
181 views

Given the differential equation, how to solve the y function with x as the independent variable?

$y\frac{dy}{dx} = x(y^4 + 2y^2 + 1)$ $y = 1$ when $x = 4$ I tired to integrate by substitution, but it doesn't seem to work out.
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0answers
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Dimensional reduction of system of ODEs

Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers: $$dX_1/dt = \ldots\\ dX_2/dt = \ldots \\ \ldots ...
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3answers
48 views

Solving the differential equation $dr=(r\cos\theta +r\sin\theta)d\theta$

$dr=(r\cos\theta +r\sin\theta)d\theta$ In my book this is under separation of variables then i tried to factor out r and divide both sides then integrate both sides but where can i find my $C_1$? I ...