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

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1
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0answers
18 views

Trying to solve nested summation problem using recurrence relation

I am trying to find a solution for $$ \phi(\lambda_s,v,m) = \sum_{r_1=1}^{\lambda_s-v} \sum_{r_2=1}^{\lambda_s-v-r_1} \cdots \sum_{r_m=1}^{\lambda_s-v-\gamma(v,m)} (1) $$ where we have $\lambda_s,v,m$...
0
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1answer
37 views

Is going from $V_{\text{L}} = L \frac{di_{\text{L}}}{dt}$ to $\frac{ V_{\text{L}} } {i_L} = L \frac{d}{dt}$ allowed?

The Laplace transform of $\frac{d}{dt} f(t)$ would be sF(s), when f(0)=0, which is something you can find in a Laplace transform table. If there is a rule that prohibits mathematical operations from $...
0
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2answers
19 views

Matrix Differentiation of $-a^T X^T y$ on $a.$

In short; what is the correct differentiation of: $$S(a)=-a^TX^Ty$$ when differentiating: $$0=\frac{∂S}{∂a}= \;?$$ Long story is; I know that: $$J(a)=\underbrace{\:\:\:a^TX^TXa\:\:\:}_u\:\...
2
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0answers
23 views

'2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
0
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0answers
9 views

Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
0
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1answer
38 views

Difficult engineering second order DE, any pointers?

I have the following engineering DE: $$rR''+R'+\alpha r(R^2_0-r^2)\lambda^2R=0$$ Where $R(r)$ is Real, $r \geq 0$, $\alpha >0$. Boundary conditions $R(R_0)=0$ and $\Big(\frac{dR}{dr}\Big)_{r=0}=...
0
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1answer
29 views

Differentiation of$ f^{-1}(x)$, where $f(x)=e^{x-1}+x^3-4x^{-3}+10$

if $f(x)=e^{x-1}+x^3-4x^{-3}+10$ then find $\frac{d(f^{-1}(x))}{dx}$ at $x=8$..... (here $f^{-1}(x)$ means inverse of $f(x)$) I was trying to solve this problem but was not able to find out the way ....
0
votes
2answers
26 views

Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
0
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1answer
18 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
0
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0answers
30 views

Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
3
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1answer
24 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
0
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1answer
28 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
1
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1answer
20 views

First principle of differentiation needs to approximate a sufficiently small integral as area?

$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$ is the solution to the differential equation $$\frac{dy}{dt} = -H(t)y$$, $H(t)$ and $y$ are scalar. However, in showing that $$y(t+\Delta t)...
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0answers
37 views

Help me solve this differential equation

I have came across where the complementary function has complex roots. Please help me solve this equation Also please use this method (operator d method ) as I know only this method I am uploading ...
0
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1answer
32 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
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0answers
19 views

Development of Calculus and differential equation question

In my post Showing propagation in ordinary differential equation, I have a hard time getting the exponential solution of propagation$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$. I now ...
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0answers
43 views

Showing propagation in ordinary differential equation

I have a very simple linear First Order Homogeneous Differential Equation with time but this makes me ponder upon the definition of derivative and integral. $$\frac{dy}{dt} = -H(t)y$$ where $y, H(t)$ ...
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4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
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0answers
7 views

If we know $h_a = H_0^{1}(x + iy),$ for the Hankel function $H_0^{(1)}$, is it possible to determine $h_b = H_0^{1}(x - iy)?$

I am wondering if there is a certain identity for Hankel functions of the first kind of order $0$. If we know $$h_a = H_0^{(1)}(x + iy),$$ where $y > 0$, is it possible to determine $$h_b = H_0^{(...
0
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0answers
18 views

Solutions to Strum-Louiville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Strum-Louiville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
0
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1answer
31 views

N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.
0
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2answers
31 views

Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
0
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1answer
31 views

How to find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency? [on hold]

Find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency.
0
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1answer
21 views

Undetermined Coeficient problem using Wolfram Alpha [on hold]

I ran this ecuation through Wolfram Alpha: y''+3y'+2y = 1/(1+e^x) Everyone in the internet agrees the answer is the one Wolfram Alpha Provides, including my teacher. However while using Variation ...
1
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0answers
23 views

Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
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0answers
41 views

Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
1
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0answers
16 views

Does there exists a notion of rate of convergence for ODE?

Suppose we wish to solve $$\dot x = f_i(x)$$ Subjected to $x(0) = x_0 \in \mathbb{R}$ where $f_i$ is some collection of continuous ($C^1$ or Lipschitz) vector field Is there an objective way to ...
0
votes
2answers
26 views

Why does quotient of basis for second order ODE solution must not equal to 0?

I'm trying to understand the concept of Second Order ODE general solution, and I need help. Why does a quotient of basis of a general solution $\neq constant$? For example: $y_1=e^x$ and $y_2=e^{-x}$ ...
0
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3answers
42 views

General chain rule help/ derivatives help.

I've been thinking too much about the chain rule and I've got myself in a muddle: Suppose $y=f(g(x))$, we can easily show that $\frac {dy}{dx} = f'(g(x))\cdot g'(x)$. I would ask please that ...
0
votes
1answer
52 views

How to find eigenvectors given complex eigenvalues

I am given that: $\vec{x}' = A\vec{x}$, where $A=\begin{pmatrix} -3 & 0 & 2 \\ 1 & -1 & 0 \\ -2 & -1 & 0 \end{pmatrix}$ I want to find the general solution of this in terms ...
2
votes
0answers
16 views

Is there a connection between generalized ODEs and stochastic ODEs

I'm working on a problem where I've run into a generalized ODE $$ \dot X \in D(X) $$ where $D(X)$ is a continuous, compact and convex subset of $\mathbf{R}^n$. To me, this problem seems in many ways ...
1
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0answers
27 views

How to solve this Sturm Liouville problem?

$\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$ Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
4
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4answers
110 views

How to solve $y''+y=x^2$?

I need to solve: $$y''+y=x^2$$ Taking the Laplace transform (and using the fact that it is a linear operator) on both sides I get: $$\mathscr{L}(y)=\frac{2}{s^3(s^2+1)}+y(0)\frac{s}{s^2+1}+y'(0)\...
0
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0answers
42 views

Solving this ODE 1

Trouble solving this ODE : $$\frac{d^2y}{dx^2}=\int_{-\infty}^{x^2/2} e^{x-t^2/2} \, \mathrm{d}t$$ $$x>0,\, y(0)=0,\, \frac{dy}{dx}(0)=0$$ in the form $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
0
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0answers
26 views

Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
2
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0answers
13 views

General Reaction Diffusion Equation Cranck Nicolson

How can i write matlab code that general reaction-diffusion equation by using Crank Nicolson Method? ∂u(x,t)/∂t= D(u(x,t))+R(u(x,t)) where D and R spatial discretizations (matrices) of linear or ...
3
votes
1answer
64 views

Solving $ \frac {dy}{dx} = \sqrt{y} x\cos(x) $ with $y(0) = 1$

I was helping someone work this problem out for an online course and I thought it'd be pretty easy since it's a first order separable DE. I ended up with $$ y = \frac {(x\sin(x) + \cos(x) + 1)^2}{4} ...
0
votes
1answer
20 views

finding solution to $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$

I tried to solve this equation with the coordinate method but I got a bit different answer compared to the suggested one by the solution manual. Where I am making the mistake in my solution? My ...
1
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0answers
23 views

finding solution to $au_x+bu_y=f(x,y)$ where $a \neq 0$

I can't get the solution in the required form of $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$$ "where g is an arbitrary function of one variable, L is the characteristic line segment from ...
1
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1answer
36 views

Show that if $T(x)=\int_0^\infty e^{-t}t^{x-1}dt $, then $T(1)=1$

I need to calculate the following Given: $ T(x)=\int_0^\infty e^{-t}t^{x-1}dt $ I need to show that $ T(1)=1 $ Solution: My logic was to plug $1$ in for $x$ before I integrated, but I am not ...
-1
votes
0answers
28 views

Definition of Laplace transform

I need to use the definition of laplace transform to determine $L(s)$ where $f(t)=e^{-t}$ on $0\leq t\leq 3$, and $2$ on $t>3$. My solution $$\begin{aligned} \int_0^3 e^{-st} e^{-t} dt + \int_3^{...
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0answers
30 views

Linear Differential Equations help [on hold]

$(y+x^3+x y^2)dx-dy = 0$ linear differential equation. help ASAP
-1
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1answer
34 views

Why is $y=h(x)$ deemed as a solution to ODE [on hold]

Can some one explain why does $y=h(x)$ deemed as a solution to a given ODE concept? it seems like an important concept, and simple too. But I must be missing some peace in this logic. Can some one ...
1
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0answers
16 views

What are the methods for solving ODEs with accuracy higher than Runge Kutta 4?

Usually, justification of using RK4 is the following: "RK4 demonstrates a better approximations than Euler and Modified Euler methods of solving ODEs and offers a good balance between accuracy and ...
0
votes
0answers
34 views

How to pass from an ODEs system to reactions?

I have the following system: $$\frac{dx}{dt}=a_1+\frac{b_1x^n}{K_1^n+x^n}-gxy-d_1x,$$ $$\frac{dy}{dt}=a_2+\frac{b_2x^m}{K_2^m+x^m}-d_2y,$$ where $a_1,a_2,b_1,b_2,K_1,K_2,g,d_1,d_2,n,m$ are real ...
3
votes
2answers
82 views

Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
1
vote
1answer
49 views

How Can solve a n order Differential Equations

How can I solve the following equetion? what is the $$h(z).$$. $$z^n (z^n+1).|h'(z)|^n=const.$$.
0
votes
0answers
16 views

Help with linearization using Taylor Series

If I sound rather clueless, it is because I am. I'm having trouble with linearizing the following non-linear system: $$ 2\frac {dy(t)} {dt} = -y(t) - 0.9u(t)³ + 1.4q(t) $$ Where u(t), q(t) are ...
1
vote
1answer
26 views

How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
0
votes
2answers
46 views

equation $u_x+u=e^{2y+x}$ (part of the solution to $u_x+u_y+u=e^{x+2y}$)

I solved/analyzed the below PDE $$\left\{\begin{matrix} u_x+u_y+u=e^{x+2y}\\ u(x,0)=0 \end{matrix}\right.$$ and have a question to the one of the steps involving the integration, see below ...