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

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3
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4answers
865 views

A 6 meter ladder…

A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My attempt:...
2
votes
1answer
115 views

Exact Differential Equations

$M(x,y)dx + N(x,y)dy=0$ is said to be a perfect differential when $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$. Let $M_y=\frac{\partial (M(x,y))}{\partial y}$ and $...
2
votes
1answer
856 views

Stability of autonomous linear systems of ODEs

Let $A$ be an $n\times n$ real matrix, and let's consider the linear system of ODEs $x'=Ax$. I'm trying to characterize the Lyapunov stability of the origin according to the real part of the ...
2
votes
1answer
350 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
2
votes
3answers
134 views

Getting equation from differential equations

I have: $\dfrac {dx} {dt}$=$-x+y$ $\dfrac {dy}{dt}$=$-x-y$ and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$ I know to do this I need to decouple the equations so that I ...
2
votes
3answers
398 views

good book to study Differential Equations throgh geometric ideas.

When studying a subject geometric intuition is important for me. The algebra books I know, do not convey such intuition. Please, recommend books with an emphasis on geometric intuition on Ordinary ...
2
votes
1answer
183 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
2
votes
2answers
510 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
2
votes
0answers
46 views

Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ...
2
votes
1answer
225 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose $(\hat{e}_r\partial_r)\...
2
votes
2answers
10k views

Complementary Solution = Homogenous solution?

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution? Let's take example $y''-3y'+2y=\cos(wx)$ and now ...
2
votes
1answer
1k views

Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.

here is the question: For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by $$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$ ...
2
votes
1answer
56 views

Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please ...
2
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1answer
210 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
1
vote
1answer
78 views

Differential equation for Harmonic Motion

Particle undergoes simple harmonic motion. Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$ Compute displacement and acceleration when the velocity is square root of $8$. ...
1
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3answers
28k views

What exactly is steady-state solution?

In solving differential equation, one encounters with steady-state solution. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t \...
1
vote
1answer
144 views

solving third-order nonlinear ordinary differential equation

I would like to solve: $(\frac{d^{2}y}{dx^{2}})^2+\frac{d^{3}y}{dx^{3}} \frac{dy}{dx}=0$ Thanks in advance.
1
vote
3answers
8k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
1
vote
1answer
138 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
1
vote
2answers
93 views

The solution of ODE $k'(x) = r(k(x))$ is infinitely differentiable if $r$ is

If there is a function $r(x)$ that is infinitely differentiable, prove that $k(x)$ is also infinitely differentiable if $k'(x) = r(k(x))$ for all $x$. I am trying to somehow prove using induction. ...
1
vote
1answer
122 views

$(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$

Hi I am trying to find the kernel of the linear differential operator $D$ $$ D\equiv\partial_{tt}+b\partial_t-a\nabla^2,\quad a,b>0. $$ We have $$ \nabla^2\equiv \frac{1}{r}\partial_r(r\partial_r)-...
1
vote
2answers
1k views

Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } \...
1
vote
1answer
225 views

Differential equations - Relation between the number of solutions and the order

The case $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$: I want to show that in the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ each differential equation has a ...
1
vote
3answers
463 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
1
vote
1answer
392 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
0
votes
2answers
2k views

`“Variation of Constant”` -method to solve linear DYs?

My school instructs to use some method called "variation of constant" (first page here) to solve linear DY more in my earlier question here. I think I solved the ...
0
votes
1answer
375 views

van der pol equation

Consider the van der Pol equation below: $(x'')+a(x^2-1)(x')+(x)=0$ I need to : Find an equilibrium point and linearize this equation near it Find solutions of the linearized equation depending on ...
0
votes
1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln u^2-nu^2](4\pi\tau)^{-n/...
0
votes
2answers
134 views

Construct the Green s function for the equation

Construct the Green s function for the equation y^''+ 2y^'+2y=0 Which boundary conditions y(0)=0 , y(π/2)=0 Is this Green s function symmetric? What is the Green s function, if the ...
10
votes
1answer
299 views

Riccati differential equation $y'=x^2+y^2$

$$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you
7
votes
1answer
196 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
7
votes
3answers
2k views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
6
votes
2answers
260 views

“Constrained” numerical solutions of ODEs with conservation laws?

I know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
4
votes
1answer
1k views

An existence of global solution of differential equation of first order

Let $f: (a,b) \times \mathbb{R} \rightarrow \mathbb{R}$ be of class $C^1$ in $D:=(a,b) \times \mathbb{R}$ and satisfies condition $$| f(t,x)| \leq A+B|x| \textrm{ for } (t,x) \in D,$$ where $A,B$ ...
4
votes
2answers
5k views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
4
votes
1answer
6k views

Series solution to $y''-xy'-y=0$

So I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I ...
4
votes
1answer
580 views

Second-order nonlinear ODE with Dirac Delta

Can anyone help me with the following differential equation? $$ 2x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0, $$ where $\delta$ represents the Dirac Delta. I tried Mathematica but with no luck. ...
4
votes
3answers
358 views

Closed form solutions of $\ddot x(t)-x(t)^n=0$

Given the ODE: $$\ddot x(t)-x(t)=0$$ the solution is: $$x(t)=C_1\exp(-t)+C_2\exp(t)$$ If we square the $x(t)$ we have: $$\ddot x(t)-x(t)^2=0$$ and the solution is given by: $$x(t)=6\wp(t+C_1;0,C_2)$$ ...
4
votes
3answers
664 views

What is an example of a second order differential equation for which it is known that there are no smooth solutions?

I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it's ...
4
votes
2answers
2k views

Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...
4
votes
2answers
161 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = 0$....
3
votes
2answers
108 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ c)...
3
votes
3answers
126 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
3
votes
2answers
247 views

Solving ODE with substitution

I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...
3
votes
1answer
535 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow \frac{1}...
3
votes
2answers
145 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
3
votes
1answer
684 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
3
votes
1answer
3k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
3
votes
1answer
176 views

Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$

Another question on ODEs, this time just wondering how I should start with this one. $$t\frac{dx}{dt}=x+\sqrt{t^2+x^2}, \qquad x(1)=0.$$ It looks like a linear ODE and so after playing around with it ...
2
votes
0answers
160 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...