Differential Definition and 1000 Threads

To be able to build a control $$ y_{1}{}'=y_1{}+y_{2} $$ $$y_{2}{}'=y_2{}+u $$ $$u \epsilon L^{2} (0,1)$$ for the care of the appropriate system solution $$y_{1}(0)=y_{2}(0)=0$$ satisfy $$y_{1}(1)=1 ,y_{2}(1)=0$$ Please kindly if you can help me Discipline is Optimal ControlHELP...

This is quite literally a showerthought; a differential equation is a statement that holds for all ##x## within a specified domain, e.g. ##f''(x) + 5f'(x) + 6f(x) = 0##. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set...

Here i added a page from my fluid dynamics book where it shows particle model for deriving the equation. My question is why pressure is more at stream side aka 'positive "s" direction'.I would expected more pressure on the other side because for example when you trying to push a rigid object or...

I'm trying to solve a differential equation of the form $$\frac{A'(x)}{A(x)}f(x,y) = \frac{B'(y)}{B(y)}$$ where prime denotes differentiation. I know that for the case ##f(x,y) = \text{constant}## we just equal each side to a same constant. Can I do that also for the case where ##f(x,y)## is not...

I am trying to deal with this problem, the question is what is the force to balance the weight W, where the rope don't have weight. The bigger pulley at the top has radius a, and the other, attached to the same axis, has radius 0.9a. The force is applied in one side of the freeling rope. I...

Please let me make questions after showing what I am studying. We first consider two particles (they may be either leptons or photons) with initial (i.e. before collision) four momentum ##p_i = (E_i, \mathbf p_i)##, ##i=1,2##. These two collide and produce ##N## final particles with momentum...

Hello, I need help deciding on whether to take ODE (MAP2302) and Calc III during the summer. Would it be wise to take ODE along with Calc III in the same semester? Some people have told me to take Calc III first because there are a few things in ODE that are taught in Calc III, but others have...

Suppose we displace the pendulum bob ##A## an angle ##\theta_0## initially, and let go. This is equivalent to giving it an initial horizontal displacement of ##X## and an initial vertical displacement of ##Y##. Let ##Y## initially be a negative number, and ##X## initially be positive. I observe...

I let ##M = 4xy + 1## and ##N = 2x^2 + \cos{(y)}##. Since ##\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}##, the equation is exact and we have $$\frac{\partial f(x,y)}{\partial x} = 4xy + 1$$ From inspection, you can tell this has to lead to $$f(x,y) = 2x^2 y + x + h(y)$$ and we...

The formula for general oscillation is: The formula for underdamping oscillation is: where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other. After some operations, we get: x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus...

Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...

In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space: dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates...

In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...

Can someone list to me (and whoever is going to view this thread) what topics in differential equations should be studied so that we can have a decent knowledge of the general physical theories in which they occur? (And I believe, they appear in all theories.) So far, I believe the two most...

I need to solve ∂2Φ/∂s2 + (1/s)*∂Φ/ds - C = 0 Where s is a radial coordinate and C is a constant. I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution? Is the method of...

253 Which of the following is the solution to the differential equation condition $$\dfrac{dy}{dx}=2\sin x$$ with the initial condition $$y(\pi)=1$$ a. $y=2\cos{x}+3$ b. $y=2\cos{x}-1$ c. $y=-2\cos{x}+3$ d. $y=-2\cos{x}+1$ e. $y=-2\cos{x}-1$ integrate $y=\displaystyle\int 2\sin...

Good evening, I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist! I am assuming a quarter car type...

I tried to derive this by myself but I'm stuck. What i did it to substitute a_{1} with a_{1} +\Delta a_{1} in the first equation, getting: (a_{1}+\Delta a_{1})\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) and trying to subtract a_{1}\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) to it. But it's not...

I'd like to understand the movement of a particle along the surface of a three dimensional graph. For example, if there is a flat two dimensional plane (z=2 for all x and y), and a unit vector describes its initial direction of movement (<sqrt(2)/2i+sqrt(2)/2j> for example), then the vector...

I got a C last semester in elementary differential equations. It was an online class using ProctorU and I always had technical difficulties, so while my homework category was a 95% my test category was around a 60%. I am a spring-semester sophomore right now and my GPA is 3.611. If I retake...

The highlighted part is what I don't understand. Due to the gate voltage increase in M1, the current in the left branch should increase. That makes sense to me. However, he then says that the voltage at node F (in other words the drain of M1) decreases? How? Look at this plot: As current in...

Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?

What is yours?

I wish to know if there is a method to work out x(t). [No matter which form f(t) is] Thank you~

on introducing a term on both sides, we have ##(x^2+xy-2xy)y^{'}=x^2+y^2-2xy## ##(x^2-xy)y^{'}=(x-y)^2## ##x(x-y)y^{'}=(x-y)^2## ##xy^{'}=(x-y)## ##y^{'}=1-y/x## ## v+x v^{'}=1-v## ...ok are the steps correct before i continue?

Hi, Could you please help me to solve a second-order differential equation given below ∂M/r∂r+∂2M/∂r2 = A [Moderator's note: Moved from a technical forum and thus no template.]

Homework Statement: first order non linear equation Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant t0=0 , T=T0 Hi, i need find an experession of T as function of t from this first order nonlinear equation: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant...

i need a simple calculation of a stirling ltd surface area calculation per watt produced and would a very rough surface and very large surface area hinder heat flow if the distance to re generator is extreme /would folding the surface area of a flat plat ltr help increase the power of engine...

The area differential ##dA## in Cartesian coordinates is ##dxdy##. The area differential ##dA## in polar coordinates is ##r dr d\theta##. How do we get from one to the other and prove that ##dxdy## is indeed equal to ##r dr d\theta##? ##dxdy=r dr d\theta## The trigonometric functions are used...

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...

1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...

In genaral relativity, how to solve differential equations is seldom be discussed. I want to know how to sole the differential equations like this: $$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$ Here ##\Gamma^i_{jk}(x)## is connection field on a manifod and ##A^i(x)## is a vector...

So this is what I have done: ##f'(t)=k*f(t)*(A-f(t))*(1-sin(\frac{pi*x}{12}))## ##\frac{1}{f(t)*(A-f(t))}=k*(1-sin(\frac{pi*x}{12}))## I see that the left can be written as this (using partial fractions): ##1/A(\frac{1}{f(t)}-\frac{1}{A-f(t)})## And then I take the integral on both sides and...

I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I...

Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably...

In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?

Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...

According to this image, in the attached files there is the demonstration of the ampere's law in differential form. Bur i have some difficulties in understanding some passages. Probably I'm not understanding how to consider those two magnetic vectors oriented and why have different name. in...

Introducing the new variables ##u## and ##v##, the chain rule gives ##\dfrac{{\partial{f}}}{{\partial{x}}}=\dfrac{{\partial{f}}}{{\partial{u}}} \dfrac{{\partial{u}}}{{\partial{x}}}+\dfrac{{\partial{f}}}{{\partial{v}}} \dfrac{{\partial{v}}}{{\partial{x}}}##...

For the first and second, I don't know if there is an analytical solution. The third I believe can only be solved with: $$ f(x,t)=c e^{\alpha \beta t}$$

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...

i am doing research to make criteria by which i can identify easily linear and non-linear and also identify its singular or not by doing simple test.please help me in this regard.

suppose y''=r^2=s y'=r 4(y''y'')-(y'y')-1=0=4(r^2)^2-(r^2)-1=4(s^2)-s-1 s=(-b±√(b^2-4ac))/2a s=(1±√17)/8 y=∫∫sdx=∫∫((1±√17)/8) dx=(1±√17)/8)(1/2)x^2+c1x+c2

I tried the substitution ##y=e^{\int z(x)}##,##z(x)## is an arbitrary function to be determined. Substitute this to the original differential equation,and dividing ##y^2## yields ##(z+1)z'+z^3+z^2+z=0##,which is a first order differential equation. Trying to solve this first order differential...