0
votes
1answer
15 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
-1
votes
1answer
24 views

Integral curves and ODE [on hold]

How do I find a first-order differential equation having the given family ofcurves as integral curves to that one: all circles through the points $(1, 1)$ and $(-1,-1)$?
0
votes
3answers
52 views

Integration and ODE

How do I integrate this? $$y'=1+\frac{y}{x}$$ I just don't know how to start. I think I gotta to try some variable changing, but I don't think I'm gonna so far with this.
0
votes
0answers
36 views

Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...
0
votes
1answer
31 views
+50

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
0
votes
1answer
41 views

Integral with square root of function of function

I have the function $y=y(x)$ with $y'=dy/dx$, and the following equation: $ky'=\pm\sqrt{k^{2}-y^{2}}$, where $k$ is constant. Integrating this, given that $y(0)=0$, should give: $y=k\sin(x/k)$. I ...
1
vote
2answers
43 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
0
votes
2answers
36 views

Why would I want to find the rate at which things were changing? Marginal cost, marginal revenue and profit

I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and ...
1
vote
1answer
34 views

Convergence of the arc length of an orbit

I was looking through this exercise but couldn't really think of a prove for it: Consider x'=f(x), with f: $\mathbb R^n \rightarrow \mathbb R^n$ continuously differentiable. Let $(p_m)_{m \in ...
0
votes
1answer
51 views

Help needed in solving a differential equation

Please help me in solving: $$a^2z+\frac{\partial^2z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=0$$ ($a$ is a constant) I plugged this in Wolfram Alpha and it outputs that this is a second ...
11
votes
10answers
1k views

What is the significance of the slope of the tangent line of a function? Why is the derivative so important?

As I finished calc 1. I can use the product rule and chain rule and resolve integrals. But I feel like its too mechanical for my taste. I know the procedure and I execute on paper without really ...
3
votes
1answer
79 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
0
votes
1answer
56 views

Differentiation of multivariable function proof

I'm looking for the differentiation of multivariable function integral $$\frac{\mathrm{d} }{\mathrm{d} x} \int_{v(x)}^{u(x)}f(t,x)dt=u'(x)f(u(x),x)-v'(x)f(v(x),x)+\int_{v(x)}^{u(x)}\frac{\partial ...
0
votes
1answer
24 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
0
votes
1answer
19 views

Solving an ODE using variations of parameters and Wronskian theorem.

So I am attempting to solve this differential equation by trying to follow an example that my professor did in class. I am just not too sure about my answer seeing as WolframAlpha gives me this: ...
0
votes
0answers
26 views

problem on partial integration in two variables

I am trying to find the solution to an assignment, the question is, to find the solution to the following fourth order differential equation in two variables; $D^4w/Dx^4+2D^4w/Dx^2Dy^2+D^4w/Dy^4=q/k$ ...
1
vote
3answers
49 views

Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
1
vote
0answers
15 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
0
votes
0answers
30 views

Equilibrium Solutions and Direction Fields

Normally I would never post my homework on stackexchange, but I have asked almost everybody that I know who could help, and they all seem at a loss. Those who remember what differential equations are ...
1
vote
2answers
50 views

Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
2
votes
1answer
40 views

Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
3
votes
2answers
189 views

On the constant of integration in solving ordinary differential equations

I very much suspect this but I'm not sure if it's correct: In solved differential equations, does the constant 'c' always represent the value of the dependent variable when the independent=0 ?
0
votes
4answers
30 views

Acceleration is given in terms of velocity. Find velocity in terms of time [closed]

Problem is from IB Math HL book from Fabio Cirrito
1
vote
0answers
40 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
6
votes
1answer
106 views

Help needed in solving a system of DE

The system of DE is: $$\frac{dI}{db}=-\frac{b}{c}\frac{dJ}{db}-\frac{2ab+1}{2c}J$$ $$\frac{dJ}{db}=\frac{b}{c}\frac{dI}{db}-\frac{2ab-1}{2c}I$$ Assume that $a$ and $c$ are constants and both $I$ and ...
4
votes
3answers
75 views

Robust Numerical ODE Solver?

I made a little explicit Runge-Kutta 4th order solver a few days ago, but when testing it against various 1st and 2nd order ODEs chosen at random (for example $d^{2}y/dt^{2} = -y \sin(y)$, ...
7
votes
5answers
382 views

Solving a separable differential equation

Solve the differential equation: $$y'=\frac{1-y^2}{1-x^2}$$ My book says the solution is: $$y=\frac{x+c}{cx+1},$$ where $c$ is a constant. It's been ten minutes I tried to verify if it was correct ...
1
vote
1answer
28 views

Differential equations with velocity of car

One may assume that as a car moves that there is a force resisting the movement that is proportional to the car’s velocity $v$. Suppose that the car has a mass of $m$ we get then that the force ...
1
vote
0answers
38 views

General solution to $f(x,y,z) = g(p,q,z)$ for all $x,y,z,p,q$ by method of inexact differential

I have a book that says that if $$ f(x,y,z) = g(p,q,z) $$ and $$ h(x,y,p,q) = 0 $$ then f and g both have the form (for instance): $$ \phi(x,y)\zeta(z) + \eta(z) $$ I'm guessing the proof has to ...
3
votes
1answer
36 views

Separation of variables method

Hi I am trying to solve the following $$\frac{dy}{dx}=\frac{2xy}{x^2-1}$$ with boundary condition $y=1$ at $x=0$. I use the method of separation of variables: $$\frac{dy}{y}=\frac{2x}{x^2-1}dx$$ ...
1
vote
1answer
31 views

Green's Functions: Strange manipulation in integral

I've been studying Green's functions and basically I've found out the book and the teacher doing some strange manipulations in integrals. Basically it has been shown the following: if we have the ...
2
votes
2answers
48 views

How to solve this first order non-linear ODE

I am struggling to solve this first order ODE. $$ u'\,^2 = a u^2 + b u^3$$ Mathematica gives me, $$ u(v) = \dfrac{a}{b} \mathrm{sech} \left( \dfrac{1}{2} \sqrt{a} \ (v + c) \right)^2 $$ So I ...
0
votes
0answers
46 views

Calculus based question

For those who knows this stuff this is probably an easy question, but I don't know in general where I should consult to learn solution of this type of problems. So, beside providing solution/hint if ...
2
votes
0answers
68 views

Solve the differential equation: $(1+\tan x)(\text{d}x-\text{d}y)+2x\ \text{d}y=0$ [closed]

$$(1+\tan x)(\text{d}x-\text{d}y)+2x\ \text{d}y=0$$ I tried a lot. Is there some rule? I tried by separating $\text{d}x$ and $\text{d}y$ but that wouldn't solve.
0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
1
vote
1answer
113 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
1
vote
2answers
57 views

integrating exponential

How can you integrate $\frac{e^{x-3}}{x^4} dx$. I tried integration by parts but it's not possible. Is some substitution possible ? I started off solving the diff eqn $(xy^2 + 3e^{x-3})dx - x^2ydy ...
1
vote
2answers
34 views

integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
4
votes
0answers
89 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
0
votes
0answers
37 views

Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...
4
votes
1answer
73 views

Find general solution of first order non-linear in a transcendental function

I have the function $$\frac{dV}{dT}=1-V^2$$ Just looking to see if my working is okay. $$dV=1-V^2dT$$ $$\frac{1}{1-V^2}dV=dT$$ Integrate $$\int{}\frac{1}{1-V^2}dV=\int{}dT$$ Let $V=\tanh(x)$ ...
0
votes
0answers
40 views

Unable to understand certain mathematical assumptions: Differential Equations

I was looking at page 49 of http://www.macs.hw.ac.uk/~bernd/F13YB1/odenotes5.pdf (page 2 of the PDF) And I came across a relatively strange argument which didn't make logical sense to me: The ...
1
vote
1answer
40 views

Finding general solution of first order DE's using integrating factor

I am asked to find the general solution of $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ I re-arrange so it is in the correct format. $$\frac{dq(t)}{dt}+\frac{1}{CR}\cdot{q(t)}=\frac{V_0}{R}$$ ...
1
vote
2answers
49 views

Find general solution of first order DE using integrating factor

I have the equation $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ And am asked to find the general solution using the integrating factor. I am a bit confused as I have been shown two ways to do it. ...
0
votes
0answers
36 views

Find the solution of this differential equation

I want to solve $\dot{\xi}(s)=\sqrt{\frac{(n-2)^2}{4}\xi(s)^2-\frac{n-2}{n}\xi(s)^{\frac{2n}{n-2}}}$ with the condition $\xi(0)=\biggl(\frac{n(n-2)}{4}\biggl)^{\frac{n-2}{4}}$. I know that ...
3
votes
3answers
50 views

How find this equation $y''+(y')^2\cdot e^x=0$

if the ODE $$y''+(y')^2\cdot e^x=0$$ such $$y(0)=1,y'(0)=1$$ Find the $y(x)=?$ my ugly methods: let $$y'=p,y''(x)=p'(x)$$ so $$p'(x)+p^2\cdot e^x=0$$ $$\dfrac{dp}{dx}=-p^2e^x$$ ...
2
votes
2answers
59 views

Laplace transform with initial value problem $y''+4y=12\sin(2t)$.

Using Laplace transforms solve the initial value problem. $$y''+4y = 12\text{sin}(2t); \qquad\qquad y(\pi)=-3, \quad y'(\pi)=-3$$ I have begun with writing: $\mathcal{L} (y'') = s^2y(s) -s y(\pi) ...
2
votes
2answers
46 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
1
vote
2answers
143 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
1
vote
3answers
39 views

1st order differential equation

I am given the following: $$ \begin{cases} x \ln x \frac{dy}{dx}+y + x = 0, &\mbox{if}\quad x>1, \\ y = 0, &\mbox{if} \quad x=e \end{cases} $$ I tried to separate it and got this: $$ -y \ ...