Differential equation Definition and 1000 Threads

Starting from equation \frac{dy}{dx}=\int^x_0 \varphi(t)dt we can write dy=dx\int^x_0 \varphi(t)dt Now I can integrate it \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^x_0\varphi(t)dt Is this correct? Or I should write it as \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^{x'}_0\varphi(t)dt Best wishes in new year...

This is the question; This is the solution; Find my approach here, ##x####\frac {dy}{dx}##=##1-y^2## →##\frac {dx}{x}##=##\frac {dy}{1-y^2}## I let ##u=1-y^2## → ##du=-2ydy##, therefore; ##\int ####\frac {dx}{x}##=##\int ####\frac {du}{-2yu}##, we know that ##y##=##\sqrt {1-u}## ##\int...

How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...

What the hell is this and is it solvable?

Hello! Consider this ODE; $$ x' = sin(t) (x+2) $$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$ x = c * e^{-cos(t)} -2 $$ and now I wanted to plug in the initial conditions and this is...

hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used. it is easy to explain nonlinearity in case of simple equation i.e when output is not...

Hello! First I tried modelling it like most mixing problems. $$ \frac{dA}{dt} = rate coming in - rate coming out $$ where dA is the volume and dt is the time rate coming in/out can be describe as; contrencation * flow rate. Now if we plug that all on $$ \frac{dA}{dt} = 35 * 0 -...

$s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?

Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...

Suppose you have a smooth parametrized path through spacetime ##x^\mu(s)##. If the path is always spacelike or always timelike (meaning that ##g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds}## always has the same sign, and is never zero), then you can define a smooth function of ##s##...

A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:- i)How many students will be...

Hello ! I need to solve this diffrential equation. $$ y^{(4)} + 2y'' + y = 0 $$ First I wanted to find the homogenous solution,so I built the characteristic polynomial ( not sure if u say it so in english as well).I did that like this $$\lambda^4 +2\lambda^2+1 = 0 $$. The solutins should be...

How to solve this first order second degree differential equation ? ##\left(\frac {dy} {dx}\right)^2 + 2x^3 \frac {dy} {dx} - 4x^2y=0 ## Thanks.

Any idea how to solve this equation: ## \ddot \sigma - p e^\sigma - q e^{2\sigma} =0 ## Or ## \frac{d^2 \sigma}{dt^2} - p e^\sigma - q e^{2\sigma} =0 ## Where p and q are constants.Thanks.

There is an mass-spring oscillator made of a spring with stiffness k and a block of mass m. The block is affected by a friction given by the equation: $$F_f = -k_f N tanh(\frac{v}{v_c})$$ ##k_f## - friction coefficient N - normal force ##v_c## - velocity tolerance. At the time ##t=0s##...

Hi everyone, Imagine I have a system of linear differential equations, e.g. the Maxwell equations. Imagine my input variables are the conductivity $\sigma$. Is it correct from the mathematical point of view to say that the electric field solution, $E$, is a function of sigma in general...

(x cos(y) + x2 +y ) dx = - (x + y2 - (x2)/2 sin y ) dy I integrated both sides 1/2x2cos(y) + 1/3 x3+xy = -xy - 1/3y3+x2cos(y) Then I get x3 + 6xy + y3 = 0 Am I doing the calculations correctly? Do I need to solve it in another way?

Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...

$\tiny{1.5.7.19}$ \nmh{157} Solve the initial value problem $y'+5y=0\quad y(0)=2$ $u(x)=exp(5)=e^{5t+c_1}$? so tried $\dfrac{1}{y}y'=-5$ $ln(y)=-5t+c_1$ apply initial values $ln(y)=-5t+ln(2)\implies ln\dfrac{y}{2}=-5t \implies \dfrac{y}{2}=e^{-5t} \implies y=2e^{-5t}$

hi guys i was trying to solve this differential equation ##\frac{d^{2}y}{dt^{2}}=-a-k*(\frac{dy}{dt})^{3}## in which it describe the motion of a vertical projectile in a cubic resisting medium , i know that this equation is separable in ##\dot{y}## but in order to solve for ##y## it becomes...

So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is: 1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system: -Here I struggle a bit with actually...

I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##

By considering a vector triangle at any point on its circular path, at angle theta from the x -axis, We can obtain that: (rw)^2 + (kV)^2 - 2(rw)(kV)cos(90 + theta) = V^2 This can be rearranged to get: (r thetadot)^2 + (kV)^2 + 2 (r* thetadot)(kV)sin theta = V^2. I know that I must somehow...

kindly note that my question or rather my only interest on this equation is how we arrive at the equation, ##v(x)=ce^{15x} - \frac {3}{17} e^{-2x}## ...is there a mistake on the textbook here? in my working i am finding, ##v(x)=-1.5e^{13x} +ke^{15x}##

now my approach is different, i just want to check that my understanding on this is correct. see my working below;

Hi, Could you please have a look on the attachment? Question 1: Why is this differential equation non-linear? Is it u=\overset{\cdot }{m} which makes it non-linear? I think one can consider x_{3} , k, and g to be constants. If it is really u=\overset{\cdot }{m} which makes it non-linear then...

ial this is the question.

>10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...

the differential equation that describes a damped Harmonic oscillator is: $$\ddot x + 2\gamma \dot x + {\omega}^2x = 0$$ where ##\gamma## and ##\omega## are constants. we can solve this homogeneous linear differential equation by guessing ##x(t) = Ae^{\alpha t}## from which we get the condition...

Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?

I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...

That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations. I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...

Well, I followed the strategy used by A.S. Parnovsky in his article (\url{http://info.ifpan.edu.pl/firststep/aw-works/fsV/parnovsky/parnovsky.pdf}) and found this differential equation: $$-\frac{g x}{C^{2}} = -\frac{\beta^{2} {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta...

for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...

Hi all, if anyone could help me solve this 2nd order differential equation, it would mean a lot. Problem: Solve the equation with y = 1, y' = 0 at t = 0 y'' - ((y')^2)/y + (2(y')^2)/y^2 - ((y')^4)/y^4 = 0 I have never solved an ODE of this kind before and I am not sure where to start...

I have the solution to the problem, and I mechanically, but not theoretically (basically, why do the C(s) and R(s) disappear?), understand how we go from ##(s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2) C(s) = (s^4 + 2s^3 + 5s^2 + s + 1) R(s)## to ##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) +...

Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...

Hi there can someone please help me with this differential equation, I'm having trouble solving it \begin{cases} y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0 \\ y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \ \text{and} \ y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big)\end{cases} \\...

Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it. I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...

I tried it but I don't know how to evaluate the integral on the last equation. Help.

I'm confused at the part how 4Vc and 48 cos(60t) are deduced, that's all.

$\tiny{27.1}$ 623 Find a general solution to the system of differential equations $\begin{array}{llrr}\displaystyle \textit{given} &y'_1=\ \ y_1+2y_2\\ &y'_2=3y_1+2y_2\\ \textit{solving } &A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}\\...

Hello forum, i want to make a samulation of a body. The body will be moved horisontal on y,x axis. I want on my simulation the body to change direction many times(for example i want to go for 10sec right and then left end right...). My question is does i need more than one differential equation...

##-\frac {dy}{dx}=\frac {3+4v}{4-3v}## ##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}## ##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}## ##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##

Greetings everyone, I am a bit new to differential equations and I am trying to solve for the natural and forced response of this equation: dx/dt+4x=2sin(3t) ; x(0)=0 Now I know that for the natural response I set the right side of the equation equal to 0, so I get dx/dt+4x=0, thus the...

I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is: \frac{\partial^2 E(z)}{\partial z^2} +...

Hello everyone! I was studying chaotic systems and therefore made some computer simulations in python. I simulated the driven damped anhatmonic oscillator. The problem I am facing is with solving the differential equation for t=0s-200s. I used numpy.linspace(0,200,timesteps) for generate a time...

Differentiating eq1 mentioned above, and using eq 2, i got : $$v\frac{dv}{d\theta}=R\frac{dv}{dt}$$ From this, i got:$$ \frac{d\theta}{dt}=\sqrt{(2/R)(g(1-cos\theta )+asin\theta)}$$ After this point, I am not able to understand what substitution or may be other method could be used to solve...

I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...

We choose an approximative solution given by $$ u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx $$ Comparing this approximative solution with the differential equation yields that $$ \frac{a_0}{2} = a $$ and the boundary conditions yields the equation system $$ a + \sum_{n=1}^N...