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

learn more… | top users | synonyms (1)

0
votes
1answer
21 views

Show that this equation together with the boundary conditions $u(0) = 2, u(\pi) = 0$ has no solution

Consider the ordinary differential equation: $u'' + u = 0$. I have no idea how to solve this, no idea what so ever. Please help.
0
votes
0answers
22 views

General solution to diffeerential equation

Given the differential equation $$\frac{dy}{dt}=\frac{4t}{1+3y^2}$$ is this the general solution? $$y+y^3=2t^2+c$$ Can we continue to simplify it?
0
votes
1answer
16 views

Verifying transport equation solution

I have just started PDE's and I have the transport equation $u_t + au_x = 0$ which has the general solution $u(x,t) = f(x - at)$ In a book I'm reading it says this can be verified by substitution ...
0
votes
0answers
8 views

Second order linear ODE arising from Euclidean heat kernel

When solving for the Euclidean heat kernel $H(t,x,y) \in C^{\infty}((0,\infty) \times \mathbb{R}^n \times \mathbb{R}^n)$, one way to proceed is to look for a solution in the form $H(t,x,y) = ...
0
votes
1answer
43 views

How to solve: y'' + 9y = sin(3t)

I need to find the particular solution to the equation: $$y'' + 9y = \sin(3t)$$ I thought we were looking for a trigonometric forcing term on the form: $$y = a\cdot\cos(3t) + b\cdot\sin(3t)$$ But ...
1
vote
1answer
18 views

Help in solving linear differential equation.

The equation is: $(xy^4 + y)dx -xdy =0$ I brought the differential terms to the same side and then divided by $y^2$ to get this. $(xy^2)dy=d(y/x)$. I tried an alternate way to simplify it which ...
0
votes
0answers
19 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
1
vote
1answer
23 views

Value(s) of the parameter $a$ that give explicit formula's

For what value(s) of the parameter $a$ is it possible to find explicit formula's (without integrals) for the solutions to $$\frac{dy}{dt}= aty +4e^{-t^2}$$ The answer is $a=-2$. I don't know how to ...
1
vote
1answer
32 views

Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm working with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
0
votes
0answers
28 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
0
votes
0answers
17 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...
2
votes
1answer
27 views

Why aren't my Laplace transform and Undetermind Coefficients answers matching up?

I might be losing my mind this morning (I am, for sure), but I can't these two techniques to give me the same answer to a basic differential equations problem. The problem is $y''-8y'+27y=0$ with the ...
0
votes
0answers
26 views

Solution of a non-linear ODE system

I'd like to find an explicit solution for the following system of ordinary differential equations: \begin{cases} \frac{dx}{dt}=-x+\frac{ax}{1+x}+\frac{by}{1+y}+c\\ \\ ...
0
votes
0answers
14 views

Stick breaking point (discretized ODE)

I cannot find nontrivial solutions to the following problem. Let $x\in[0,1]$ and $y(x)$ be the deflection of the stick. Then this is described by the diff.eq.: $$\alpha^{-1} P y(x)+y(x)''=0 $$ where ...
0
votes
0answers
12 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
1
vote
1answer
23 views

Decide the smooth function $r : \mathbb R \rightarrow \mathbb R$ of the equation $r(t)^2 + r'(t)^2 = 1$.

Suppose $r:\mathbb R \rightarrow \mathbb R$ is a smooth function and suppose $r(t)^2 + r'(t)^2 = 1$. I want to determine the function $r(t)$. I see that $r(t)^2 + r'(t)^2 = 1$, so I could take $r(t) ...
1
vote
1answer
44 views

Is there a unique solution of $\gamma(t)= f''(t)f(t) $ with $f(0) =0$ and $f'(0)=1$?

Consider, \begin{align*} \gamma(t) &= f''(t)f(t) \\ f(0) &=0 \quad f'(0)=1 \end{align*} where $f(t)$ is an unknown function and $\gamma(t)$ is a known function. Is there a unique solution ...
-1
votes
0answers
12 views

diffusion equation [on hold]

I'm kinda lost with this problem. I don't know how to solve it. If somebody can help me I will be so thankfully. I'm so confuse.If somebody know a reference problem that would help a lot
1
vote
0answers
18 views
0
votes
0answers
24 views

heat equation, total heat energy [duplicate]

I'm having a hard time with this problem. I get the situation, but I just don't know how to model it and show part b and part c. I will be so thankfully.
0
votes
1answer
24 views

Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
-1
votes
0answers
25 views

How to solve the vector differential equation? [on hold]

I'm new to this section, so I'm trying to solve vector differential equations, and I need some guidance. Could anybody give a step-by-step process for doing so, so that I could do some more problems ...
0
votes
3answers
28 views

Intro to Differential Equations Problem

Show that $y(t)= C_1 e^{2t} + C_1 e^{-2t}$ is a solution to the differential equation $y'' - 4y = 0$. $C_1$ and $C_2$ are arbitrary constants. This was the first part of the problem which I ...
2
votes
1answer
32 views

Solving $r^2 u_{rr} + 2ru_{r} + r^{2}u = 0$ directly

The problem I am working on boils to solve the differential equation $$r^{2}u_{rr} + 2ru_{r} + r^{2}u = 0.$$ The solution to this equation is the spherical Bessel function $u(r) = \sin(r)/r$. However, ...
1
vote
2answers
17 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
1
vote
1answer
22 views

Methods of Solving Ordinary Differential Equations - A Small Question

I've spent some weeks now trying to learn how to solve ordinary differential equations, and I am now studying the Laplace transform and how this can be applied to solve ODEs. I feel a little bit ...
1
vote
0answers
16 views

Solving a DE with no initial conditions

I'm having some sort of difficulty on my signals homework. I am given the following problem. Where u(t) is a unit step function. For whatever reason, most of the problems assigned have no initial ...
0
votes
1answer
39 views

Showing flows converge in the phase plane

I have a system of ODEs: $$\dot{x} = \frac{m_1 x (1-x-y)}{a_1 + 1-x-y} - x$$ $$\dot{y} = \frac{m_2 y (1-x-y)}{a_2 + 1-x-y} - y$$ I'm trying to show that all the flows converge to the point ...
3
votes
2answers
45 views

How to solve the following linear differential equation?

I'm having trouble solving the following differential equation: $y'(x)=\frac{8A^2x}{(1+4A^2x^2)^2}\cdot y-4Bx$ $A$ and $B$ are real constants. I would be very grateful for any help. Thanks in ...
0
votes
1answer
19 views

Show $y_1(t) = y(t)\int^t_{t_0} \frac 1 {x_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$

Suppose $(I,y)$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$. Assume $y(t) \neq 0$ for $t \in I$ and let $t_o \in I$. I want to show that $(I, y_1)$ where $$y_1(t) = y(t)\int^t_{t_0} \frac ...
2
votes
1answer
21 views

Eigenfunctions of the laplacian (1 dimension)

I have the following problem: $\frac{d^2 u}{dx^2}(x)+\lambda u(x)=0, x \in (a,b)$ and $u(a)=u(b)=0$. The general solution (for $\lambda>0$) is $u(x)=c_1\cos(\sqrt\lambda x)+c_2 \sin (\sqrt\lambda ...
0
votes
2answers
19 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
1
vote
0answers
11 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
0
votes
0answers
20 views

Verify that $e^{at}$ is the only solution to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$

Find all solutions to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$ By inspection, I see that $e^{at}$ is a valid solution. However, my problem is to verify that $e^{at}$ is the only ...
0
votes
1answer
57 views

Why is -ln x is not equal to 1/ln x?

I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that -ln x is not equal to 1 / ln x. I ...
-2
votes
0answers
30 views

solve $y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$ [on hold]

Let $y:[0,\infty) \rightarrow \mathbb{R}$ be twice continuously differentiable and satisfy $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$$ then which of the following is true 1. ...
0
votes
1answer
84 views

Solving a first order linear ODE and determining the behavior of its solutions

(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$ be the value ...
3
votes
1answer
46 views

Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
1
vote
0answers
24 views

what is degree of given PDE…

When the given differential eqn is completely free from radicals the the final exponent on the highest order derivative amounts degree of given differential eqn. In present case it is 3 or 6. i.e. ...
0
votes
1answer
48 views

Set up differential equation

As people get older, they perceive time differently. The older one is, the faster time goes by. To quantify this issue, we create a model: The entire perceived period of time shall be $w(t)$ . A ...
0
votes
2answers
32 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
0
votes
0answers
22 views

Properties of the solution of an initial value problem

I have an IVP which can not be solved for explicitly of the form: $y''(t) = f(y)(y')^2 +ay'-g(y)$ $y(0)=0, y'(x_1) = h(y)>0$ with $y,y',y'' \in \mathbb{R}_+$, $x \in [0,x_1]$ and I know that the ...
1
vote
1answer
42 views

Given a solution of a differential equation, determine the differential eqution itself

Sorry if my layout is bad, I'm new. So this question was asked a couple of years ago on an exam about differential-equations. Suppose you have a third order differential-equation with the following ...
3
votes
2answers
35 views

How to solve differential equation $\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$

My task is to solve for $y$ from: $$\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$$ I have been given the answer, but I would like to calculate this myself also. $\lambda$ is a ...
0
votes
1answer
37 views

differential equation degree doubt

$dy/dx = sin^{-1} (y)$ this is a form of $dy/dx = f(y).$ so degree should be $1.$ but if i write it as $y = \sin(dy/dx).$ then degree is not defined as it is not a polynomial in $dy/dx $ please ...
0
votes
1answer
65 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
0
votes
0answers
19 views

monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
2
votes
1answer
24 views

the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$

Find all possible $(\lambda,u)$ where $\lambda \in \mathbb R$ and $u\ne0$, to the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$ My Effort: for ...
0
votes
0answers
24 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
1
vote
1answer
26 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...