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
3 views

ODE x'' + 2x' +5x, x(0)=1,x'(0)=-1, Solve using Laplace Transform

I got to X=$$\frac{\left(\:s^3+s^2+9s+12\right)}{\left(s^2+9\right)\left(s^2-2s+5^{\:}\right)}$$ Is it correct, if yes how do i find the inverse of it?
3
votes
0answers
22 views

Poisson Process Derivation.

I was looking at a derivation for the poisson process , which tells the number of events occurring in time $t$ , I came across the following differential equation : $\frac{d}{dt}(P_n(t))$ = ...
-1
votes
2answers
48 views

Do the initial value problem $dy/dx=2y^{1/2}$ , y(0)=a has infinitely many solution for a=0.

please solve the initial value problem $dy/dx=2y^{1/2}$, y(0)=a. I wanted to know that this problem admits infinitely many solutions for a=0 or admits infinitely many solutions for $a\geq 0$
2
votes
1answer
22 views

Basic Ordinary Differential Equation Help

I have the following $3$rd order, non-linear, homogeneous differential equation: $$y''' + ey'' + y' + (d + e)y - dy^2 = 0,$$ where $d,e$ are constants. My questions: Have I classified this ...
0
votes
2answers
37 views

Solving differential equation , Calculate concentration of sands in water?

$V=$ volume of water in the water tank $M(t) =$ mass of sands in the water at time $t$ $K(t) =$ Concentration of sands in water at time $t$ $R=$ rate of water flowing out Concentration sands in ...
3
votes
2answers
42 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
0
votes
0answers
10 views

PDE reduced to ODE Uniqueness??

Could you please help me with the following problem. As a first help, I know the solution of the following ODE: \begin{align} j_1(t)[r \log(j_1(t)) + \beta] &= j_1'(t) \\ \nonumber j_1(T) ...
-1
votes
0answers
12 views

Quantative Methods for Business question [on hold]

Economists theorize that the recent recession has affected men more than women because men are typically employed in industries that have been hit hardest by the recession.​ Women, on the other​ hand, ...
1
vote
0answers
16 views

Existence and uniqueness solution of a differential equation

If I have the following equation: $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'e^{\int_0^t\int_0^1G(|r-r'|)y(s,r')dr'ds}-y(t,r)$ $ y(0,r)=a(r)$ where $G:\mathbb{R}^+\to\mathbb{R}$ is ...
0
votes
0answers
36 views

More elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$

Is there a more elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$ than to use Euler's identity and get the general solution through brute computation?
3
votes
1answer
26 views

Trajectories that connect equilibrium points

Suppose I consider the autonomous system \begin{align*} x' &= F(x, y)\\ y' &= G(x, y) \end{align*} where $F$ and $G$ are nonlinear and my task is to draw the phase portrait of the above ...
1
vote
1answer
42 views

Solve the differential equation $\{xy\log \frac{x}{y}\}dx+\{y^2-x^2 \log \frac{x}{y}\}dy=0$ given that $y(1)=0$.

Solve the differential equation $\{xy\log \frac{x}{y}\}dx+\{y^2-x^2 \log \frac{x}{y}\}dy=0$ given that $y(1)=0$. I was able to find the solution to it by the method of solving homogeneous ...
0
votes
0answers
17 views

Does solution to particular system of O.D.E. with boundary conditions exist and unique?

I am dealing with the following system of $2$nd order differential equations: $$\ddot{y_k}+\frac{\ddot{y}_{N,k}}{2}+a\dot{y}_{N,k} = \gamma_k^2 y_k$$ where $y_{N,k} = \sum_{i=1,i\not=k}^{N} y_i ...
0
votes
1answer
36 views

Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
1
vote
1answer
20 views

Changing a heaviside function into a one line function

$$h(t) = \left\{\begin{array}{l}1,\, \pi\leq t<2\pi\\ 0,\, 0\leq t<\pi\text{ and }t\geq2\pi\end{array}\right.$$ I need to change $h(t)$ into a one line function. I believe it to be ...
1
vote
1answer
19 views

How to calculate Z when doing Bernoulli differential equation?

I'm just learning how to do a Bernoulli differential equation and I'm stuck at the part where you have to use Z (others call it U). For example: When (y^-3)y' + (1/2x)y^-2 = -(1/2)X² * sin²x*cosx ...
1
vote
0answers
30 views

2nd Order Nonhomogeneous with varying coefficients

Is there a way to solve or get an analytical approximation to this equation? $z''(t) + z'(t)\frac{(\omega_0 + \Delta\omega (1 - e^{-\frac{t}{\tau }}))}{Q} + z(t)(\omega_0 + \Delta\omega (1 - ...
1
vote
2answers
44 views

Fitting driven Harmonic Oscillator

I've got some datapoints of a turning disc. It is supposed to obby the following differential equation: $I\ddot{\theta}+\gamma\dot{\theta}+k\theta=\tau$, So it should have the form of a driven ...
0
votes
1answer
26 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
5
votes
0answers
34 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
0
votes
1answer
36 views

Solve differential equation

How can we solve (if a closed form expression for f(x) can be found) the following first-order linear differential equation? $$f'(x)=f(x)\cdot (\cos x+\tan x)$$ I have found that one function which ...
2
votes
2answers
52 views

Solving $y' + \frac{1}{2}xy + y^{2} = 0$

I am trying to solve the ODE $$y' + \frac{1}{2}xy + y^{2} = 0.$$ Mathematica gives that the answer is $$y(x) = \frac{e^{-x^2/4}}{C + 2\int_{0}^{x/2}e^{-t^{2}}\, dt}.$$ Of course, if I take this answer ...
1
vote
2answers
18 views

Quadratic equation with several variables

How does $$y^{2} - 4y -t^{2} - C = 0$$ Become $$y = 2 \pm \sqrt{t^{2} +2C + 4}$$ I know its the quadratic formula but I dont know how it got it that point The original equation is $$\frac{dy}{dt} ...
3
votes
3answers
47 views

Laplace Transform of a Heaviside function

Find the Laplace transform. $$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$ I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$ Finding $F(s)$ is the hard part for me. My ...
3
votes
4answers
91 views

Simple differential equation( introduction but need some basic explanation)

I have a couple of questions before I dig deeper into my calculus book. First: I have learned that $\frac{d}{dx}\frac{x}{y}$=$\frac{y x'-x y'}{y^2}$ never really gotten a proper explanation for ...
1
vote
2answers
72 views

Is $ \cos² y = 0 $ a solution?

I'm studying math for school. We're solving separable differential equations. One of the exercises is: $$ \frac{\Bbb d y}{\Bbb d x} = \frac{ (\cos y)^2 \tan y }{1+x²}$$ If you separate the ...
0
votes
0answers
18 views

How to find the matrix associated with the differential equation? [on hold]

How to solve ordinary differential equations using matrices.
1
vote
0answers
21 views

Is there a physical meaning of ranking in differential algebra?

The main stone in the Ritt's Algorithm from differential algebra is ranking. If we consider an example of a differential polynomial with two variables $x$ and $y$. Then how can we say $x$ is ranked ...
1
vote
2answers
48 views

Differential equation: $Ay'' + By' + Cy = h(x)$

I'm stuck solving the equation $y'' - 3y' + 2y = 2x^3-30$. The auxiliary equation is $k^2 - 3k + 2 = 0$ where $k_1 = 1, k_2=3$. Thus the general solution is: $$y_g = C_1e^x + C_2e^{3x}$$ Then, I ...
1
vote
3answers
38 views

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 ...
3
votes
1answer
55 views

PDE: solving Fokker-Planck equation with initial and boundary condition

Here is the problem. We have the following simple PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - a\frac{\partial p(x,t)}{\partial x} + \frac{D}{2} \frac{ \partial^2 p(x,t) }{\partial ...
0
votes
1answer
29 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 : ...
0
votes
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. ...
1
vote
0answers
13 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
votes
2answers
52 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?
1
vote
2answers
61 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
votes
1answer
62 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; $$x(0) = 0$$ ...
0
votes
1answer
25 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 ...
4
votes
1answer
39 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
votes
1answer
32 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
votes
0answers
71 views

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 ...
-1
votes
0answers
33 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 ...
1
vote
0answers
17 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) ...
-1
votes
0answers
11 views

Qualitative Ordinary differential equations [closed]

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
2
votes
3answers
37 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
votes
1answer
41 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 ...
0
votes
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 ...
1
vote
1answer
26 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
votes
1answer
43 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
vote
1answer
17 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 ...