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

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Proof of the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann ...
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1answer
54 views

Transforming ODEs into exact equations.

I want some examples of ODEs that can only be solved by transforming them into exact equations. They shouldn't be solvable with; Direct integration, separation of variables, manipulating a reverse ...
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1answer
29 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
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1answer
29 views

The system $\dot{x}=x^2$, $\dot y=-y$, has infinitely many (local) center manifolds

Consider the system, \begin{align} \dot{x}&=x^2 \\ \dot y&=-y \end{align} I am trying to show that this system has infinitely many local center manifolds. Here is what I have done so far: ...
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0answers
26 views

Symbolic solution to a nonlinear ordinary differential equation problem

Suppose $y=y(x)$ is infinite continuous in $\mathbb{R}$, and $y(-1)=0$, how can we obtain the analytic solution in closed form to the following nonlinear ordinary differential equation: $$ ...
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27 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
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3answers
44 views

Differentiate the function $F(x)=\left(\int_0^x te^tdt\right)^6$ [on hold]

If $$F(x)=\left(\int_0^x te^tdt\right)^6$$ what is $F'(x)$?
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1answer
35 views

RL circuit as a system of first-order ODEs

The system is as follows:\begin{align}i_1&=i_2+i_3,\\50\sin t&=6i_1+i_2'+5i_2,\\50\sin t&=6i_1+i_3',\end{align} I have to find $i_2,i_3$. This is my first circuit I'm trying to solve, but ...
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1answer
17 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
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1answer
36 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
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3answers
51 views

Why is $ A_1 x + … + A_n x^n $ a solution of $ \sum_0^{n} (-1)^n \frac{x^n}{n!} \frac{d^n y}{d x^n} = 0 $?

I was playing(/fiddling) around with some maths and I saw this pattern( where $ A_n $ is a constant.): $ A_1 x $ is a soultion of: $$ \frac{y}{x} - \frac{dy}{dx} = 0 $$ $ A_1 x + A_2 x^2 $ is a ...
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0answers
25 views

Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now ...
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1answer
34 views

Derivation of Lagrange-Charpit Equations

I am working through the derivation of the Lagrange-Charpit equations presented in this Wikipedia article: http://en.wikipedia.org/wiki/Method_of_characteristics#Fully_nonlinear_case I am interested ...
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2answers
44 views

First order ODE: $tx'(x'+2)=x$

$$tx'(x'+2)=x$$ First I multiplied it: $$t(x')^2+2tx'=x$$ Then differentiated both sides: $$(x')^2+2tx'x''+2tx''+x'=0$$ substituted $p=x'$ and rewrote it as a multiplication $$(2p't+p)(p+1)=0$$ So ...
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3answers
50 views

Solve $y' = x^4y+x^4y^4$

Solve the differential equation $$y' = x^4y+x^4y^4.$$ I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one. ...
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2answers
66 views

Initial value problem for 2nd order ODE $y''+ 4y = 8x$

How can I go about solving this equation $y''+ 4y = 8x$? Progress I found the general solution for its homogeneous form. What I don't know is how to find its particular solution.
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3answers
56 views

First order ODE: $y^2+2yy'x+2xy'+y=0$

$$y^2+2yy'x+2xy'+y=0$$ I really have no idea how to do this, I cant fit it into any of the schemes I already know. Also nothing factors out. Maybe should I try differentiating both sides?
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1answer
25 views

the jump in $\ddot y$, Laplace transform

Given the following IVP: $$\ddot y+4y=\cos t-\cos t \cdot \theta(t-2\pi), y(0)=0, \dot y(0)=1$$ Check that $y(t)$ is continuous at $t=2\pi$. Find the jump in $\ddot y(t)$ at $t=2\pi$ i.e find $\lim ...
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1answer
27 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
2
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1answer
39 views

Solving the differential equation $9x(1-x)y''-12y'+4y=0$

Solve in series the following ODE: $$9x(1-x)y''-12y'+4y=0$$ expanding $y(x)$ about $x_0=0$. My guess: I think it is by Frobenius series since it is not an ordinary point.
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1answer
55 views

What is wrong with this separation of variables?

I know a number of ways of solving this basic DE: $\ddot{u} = -u$ Besides the fact that the solution is obvious, one can do: $\ddot{u} = \frac{d\dot{u}}{dt} = \frac{d\dot{u}}{du}\frac{du}{dt} = ...
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0answers
13 views

How the Jacobian is connected to the movement of particle from one domain to another? [on hold]

I am dealing with the proof of Reynold-Transport Theorem. There the Jacobian is used for the changing position of particles from one domain to another. Can anyone help me to understand what does ...
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5answers
83 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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1answer
31 views

Solution of $y''(x) -k = \delta(x-x_0)y(x)$

I need to solve following differential equation $y''(x) -k = y\delta(x-x_0)$ subject to boundary conditions \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} I am not sure if it is possible ...
2
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0answers
31 views

second order differential equation with Green's function

I need to solve following differential equation \begin{eqnarray} y''(x) - k = \delta(x-x_0) \end{eqnarray} subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} Is it ...
2
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1answer
27 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
2
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1answer
26 views

Finding a solution basis

Find a real solution basis of $$y'=\left( \begin{matrix}-1&-2&0\\0&2&0\\-1&-3&2\\ \end{matrix} \right)y.$$ The characteristic equation of this matrix is $$P(t) = ...
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2answers
24 views

Initial value problem with a delta term

Im having trouble solving this initial value problem. I know how to solve it without the delta-term (C1*e^(lambda*t)*S1 + C2*e^(lambda*t)*S2), but how do i solve it with a delta term?
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1answer
44 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
2
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1answer
49 views

Confused over the solution of partial differential equation $xu_x+u_t=0$

Consider, $$ \displaystyle x\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t} = 0 $$ with initial values $ t = 0 : \ u(x, 0) = f(x) $ and calculate the solution $ u(x,t) $ of the above ...
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4answers
52 views

Differential equation with bounded solutions

What are the possible values of $c,d\in\mathbb{R}$ such that any $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f''(x)+f'(x)+cf(x)-dx=0$ is bounded? My approach was consider $c=0$ which give that for any ...
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0answers
24 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
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0answers
11 views

Computer Code Friendly Books On Differential Equations?

When I need a differential equation for this or that application I generally search (by hand) through old paper and ink books written by mechanical or electronical (electrical) engineers. Sometimes my ...
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1answer
77 views

Show f is not differentiable at x=0

(c) {22 markes} Let $$ f({\bf x}) = \begin{cases} \dfrac{x_1 x_2^2}{x_1^2 + x_2^2} & : {\bf x} \ne {\bf 0} \\[1ex] 0 & : {\bf x} = {\bf 0} \end{cases} ...
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2answers
43 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
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3answers
57 views

Finding a curve - first order ODE

Find a curve whose tangent lines create a triangle of area $2a^2$ with the $x-$ and $y$-axes. First off, the tangent line equation is:$$y=f'(X)(x-X)+Y$$ where $(X,Y)$ is an arbitrary point. I ...
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0answers
31 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
3
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5answers
54 views

Solve $y''-3y'+2y=x^2$

Solve $$y''-3y'+2y=x^2$$ My approach: Homogen solution: $$y = Ae^x +Be^{2x}$$ Particular solution: $$ y_p = x(Ax^2+Bx+C) = Ax^3+Bx^2+Cx $$ $$ y_p' = 3Ax^2 + 2Bx + C$$ $$y_p'' = 6Ax + 2B$$ Put his ...
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2answers
76 views

Is $\dfrac {dy} {dx} = \dfrac {2x} {3y}$ a homogeneous differential equation?

I have a differential equation $\dfrac {dy} {dx} = \dfrac {2x} {3y}$ whose solutions are $y = \pm \sqrt {\dfrac 2 3}x $ which when I back-substitute I get $LHS=RHS$. From the definition on ...
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1answer
35 views

Is this Differential equation a linear DE?

$\Large{y\frac{dy}{dx}-xy=0}$ Could you please explain to me why it is or isn't? Much obliged, thank you.
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3answers
100 views

Finding 2nd solution of second order ODE

Function $x_1(t)=e^t$ is a solution of: $$tx''-(2t+1)x'+(t+1)x=0$$ Find second lineary independent solution. I tried to do it the usual way, substituting $x=e^{\alpha t}$. But that gives me the only ...
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0answers
11 views

An ordinary differential equation involving Torricelli's law

$\textbf{Problem}$. Torricelli's law states that (under certain ideal circumstances) fluid will leak out of a hole at the base of a container at a rate proportional to the square root of the height of ...
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1answer
14 views

Showing the following differential equation is exact

I'm asked to show that the attached differential equation is exact: link. I know I have to show that Nx=My. In this particular equation, M = -x/siny - 2 and N = ((x^2+1)cosy)/(1-cos2y), and all I ...
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1answer
27 views

First Order Linear Differential Equation with $ P(x)=0$

A first order linear differential equation can be written as: $$\frac{dy}{dx} + P(x) y = Q(x)$$ If $P(x)$ here is equal to zero, will the differential equation still be linear?
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1answer
39 views

Solving $y'' - y' -2y=0$

$$y'' - y' -2y=0$$ I've a solution that believe is wrong. I also have the correct solution, so this isn't a question about how to solve the above equation, but my question is, what's wrong with this ...
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3answers
64 views

explicit solution for second order homogeneous linear differential equation

If we consider the equation $(1-x^2)\dfrac{d^2y}{dx^2} -2x \dfrac{dy}{dx}+2y=0, \quad -1<x<1$ how can we find the explicit solution, what should be the method for solution?
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0answers
31 views

Transpose/multiplication of 3D matrices

I have $A(p)=\begin{bmatrix}p_1 &p_2 & p_3\\ 2p_1 &2p_2^2 & 4p_3^3\\ 3p_1 &3p_1 & 10\\ \end{bmatrix}\tag 1$ $ p= {\left(\begin{array}{c}p_1\\p_2\\p_3\\p_4 ...
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1answer
68 views

Non linear ordinary differential equation

How to solve the ordinary differential equation $\frac{d^2y}{dx^2}+\sin(x+y)=\sin x,y(0)=0,y'(0)=1$ Then its possible to solve it by numerical methods?
2
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1answer
90 views

Interval of existence of a certain first-order ODE [duplicate]

Without solving the following initial value problem, determine the interval in which the solution is certain to exist: $$\dfrac{dy}{dx}+(\tan x)y=\sin x, \ \ \ y \left (\frac{\pi}{4} \right )=0.$$ ...
3
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1answer
40 views

Find a solution to $y''-4y'-5y=2e^{2t}$ using variation of parameters [on hold]

Can anyone help me solve this problem? Find a solution to $y''-4y'-5y=2e^{2t}$ using variation of parameters.