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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1answer
26 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 ...
2
votes
1answer
201 views

Sturm-Liouville Eigenvalue Question

Consider the regular Sturm-Liouville Problem: $$-\frac{d}{dx} \Bigg( p(x)\frac{dv}{dx} \Bigg)=\lambda \rho (x)v$$ $$\alpha _1v(0)-\beta _1v'(0)=0$$ $$\alpha _2v(L)-\beta _2v'(L)=0$$ with ...
0
votes
1answer
22 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 ...
1
vote
3answers
55 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 ...
3
votes
0answers
149 views
+100

First order differential equation involving inverse function

I am wondering if there is a way to solve a differential equation of the following form: $$\frac{d f(x)}{dx} = g(x)\left(\frac{1}{g(f^{-1}(x))} + \frac{1}{k}\right)$$ We can assume that $f(x): [0,T] ...
2
votes
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 ...
1
vote
1answer
24 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 ...
2
votes
0answers
21 views

$\dot{x}=x^2, \quad \dot y=-y$. This system 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: ...
1
vote
0answers
24 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: $$ ...
2
votes
1answer
31 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 ...
0
votes
0answers
25 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 ...
2
votes
3answers
42 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 $F'(x)$?
0
votes
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] ...
2
votes
3answers
50 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 ...
1
vote
1answer
35 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 $\frac{dy}{dx} = .75 -.005y$ ...
2
votes
4answers
196 views

how to draw graphs of ODE's

In order to solve this question How to calculate $\omega$-limits I'm trying to learn how to draw graphs of ODE's. For example, let $p\in \mathbb R^2$ in the case of the field $Y=(Y_1, Y_2)$, given by: ...
0
votes
1answer
36 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
5
votes
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 ...
3
votes
3answers
55 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?
0
votes
5answers
82 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 ...
1
vote
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 ...
1
vote
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. ...
4
votes
1answer
218 views

PDEs with non-local terms

Not sure if I've used the correct terminology here (`non-local'). I think the lack of knowing the correct terminology is why I haven't been able to find any information about my query thus far. I'm ...
3
votes
2answers
59 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.
1
vote
0answers
49 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
2
votes
1answer
94 views

Closed form of the solution of a nonlinear differential equation

I should solve the following problem: given a function $u(x)$, the sum of the function and its reciprocal must be equal to the integral of the function raised to $k$. Taking the derivative of the two ...
6
votes
0answers
147 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
1
vote
1answer
34 views

Finite-Dimensional Subspaces Invariant under Differentiation

Let $X$ be the linear space of complex continuously-differentiable functions on $\mathbb{R}$. If $M$ is a non-trivial finite-dimensional subspace of $X$ which is invariant under differentiation, does ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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} ...
2
votes
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.
1
vote
1answer
129 views
+100

Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
2
votes
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} = ...
8
votes
3answers
307 views

Why can't I solve this homogenous second order differential equation?

I've been banging my head on the wall for quite some time trying to come up with a solution to the following: $$\frac {\partial^2 y(x)} {\partial x^2} + (A-B*V(x)) y(x) = 0 $$ $$V(x) = (36 + (2 - ...
0
votes
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 ...
0
votes
2answers
42 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 ...
2
votes
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
votes
1answer
30 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
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
2answers
23 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?
2
votes
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) = ...
2
votes
2answers
198 views

Third order ODE initial value problem,solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$ ???

$y''' + y'' -y' -y=0$ $y(0)=7,y'(0)=-3,y''(0)=\alpha$ Find all values of $\alpha$ for which the solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$ Here is my work I used the cubic ...
1
vote
3answers
50 views

Solving an ordinary differential equation with initial conditions

Can someone please help me with this ODE problem? Here is the question: Consider the ODE $ {d^2 U\over dx^2} - [{s^2\over c^2}]U=e^{{-sx\over v}}. U(0) = 0, U(x)$ is bounded as $x$ goes to ...
10
votes
3answers
11k views

Definition of a Differential Equation?

Here is one definition of a differential equation: "An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ...
0
votes
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: ...
1
vote
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 ...