What is Differential eqautions: Definition and 94 Discussions

Finding the first and second derivative of out ansatz, $$\dot x(t)=A(cos(\omega_0 t) - t\omega_0 sin(\omega_0 t)) + B(sin(\omega_0 t) + t\omega_0 cos(\omega_0 t))$$ $$\ddot x= A(-2\omega_0 sin(\omega_0 t) - t{\omega_0}^2cos(\omega_0 t)) + B(2\omega_0 cos(\omega_0 cos(\omega_0 t)...

Why the navier-stokes equation don't have a solution ?

Here is only my solution: ##A_1 \frac{\mathrm d h}{\mathrm d t}=-A_2\sqrt{2hg}##, so by integrating we get ##h(t)=\left(\sqrt{h_0}-\frac{A_2}{2A_1}\sqrt{2g} t\right)^2.## Setting ##h(T)=0## we get ##T=\frac{A_1}{A_2}\sqrt{\frac{2h_0}{g}}.## By doing the first time derivative of ##h## we...

Since energy is conserved and the particle is initially at rest, we can determine that ##E(0) = m_0 c^2##, so $$ m_0 c^2 = \sqrt{ c^2 p^2 + m_0^2 c^4 } + \alpha x. $$ Squaring this eqation gives $$ m_0^2 c^4 = \alpha^2 x^2 + c^2 p^2 + m_0^2 c^4 + 2 \alpha x \sqrt{ c^2 p^2 + m_0^2 c^4 }...

If I have been given a system of inhomogeneous linear ODEs, $$ \vec{x'} = \begin{bmatrix} 4 & -1 \\ 5 & -2 \\ \end{bmatrix} \vec{x} + \begin{bmatrix} 18e^{2t} \\ 30e^{2t}\\ \end{bmatrix} $$ I have found its particular solution to be: $$ 1/4 \begin{bmatrix} -31e^{2t} - 25e^{6t} \\ 85e^{2t} -...

We have to solve $$ \begin{align*} y'' - 5y' - 6y = e^{3x} \\ y(0) = 2,~~ y'(0) = 1 \\ \end{align*} $$ Applying Laplace Transform the equation $$ \begin{align*} L [ y''] - 5 L[y'] - 6 L[y] = L [ e^{3x} ] \\ s^2 Y(s) - \left( s y(0) + y'(0) \right) - 5s Y(s) + y(0) - 6 Y(s) = \frac{1}{s-3} \\...

Hello : Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes Best regards HB

The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...

(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?

I will be taking intermediate mechanics next semester, and am a bit concerned about potential gaps in my mathematical knowledge. Long story short, I used to be a physics major, switched to electrical engineering, and then decided to double major after a semester in EE. The issue is that, as a...

Namely, in the system, I have obtained the value of parameters L, M, A and D, because I treat the other organism as equal to zero, i.e., it doesn't exist, but I am struggling about the values of B and C, that are coupled with the product of x and y. Can anyone help me how to obtain those values...

One thing that is given in paper (attached) is a operating set point for temperature which is given as 20 for day and 16 for night but I do not know whether its initial condition for temperature or not. Can anyone please guide me that what kind of equation is it and how can I solve it with these...

I was dealing with nonlinear systems of differential equations like the Lorenz equations (https://en.wikipedia.org/wiki/Lorenz_system). Now there is a trapping region of this system defined by the ellipsoid ρx^2+σy^2+σ(z-2ρ)^2<R. I wondered how this region is found and I found out that a...

Hi, I am an undergrad looking to purchase a good textbook on differentials for my course which I will be taking soon, and the textbook listed for the differentials course is this one (https://www.amazon.com/gp/product/1118531779/?tag=pfamazon01-20) which apparently is not very good. So can...

Ben Sparks uses https://www.GeoGebra.org to explain (and code) the so-called SIR Model being used to predict the spread of cornavirus (COVID-19). A pretty cool way to visualize a set of differential equations.

My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...

From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.## Let ##f(t) = 2e^{3t-30}##. Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...

I know how to solve ODEs using both methods. The problem I'm having is knowing when to use one and not the other. If someone could help clarify this for me. I can't find the correct section in my textbook.

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...

Summary: Mechanics problem related with Calculus (differential equations) Hi everyone, I would like some help in that task, if anyone would be willing to help :) Namely I have a problem from particle dynamics. "D:" means given info... so, D: m,g,h,b, miu. We're looking for v0 and S as given...

. Above is the figure of the problem. I am trying to solve x(t) and differentiate it to obtain v(t); however, I have difficulty solving the differential equation shown below. $$ v(t)=\int a(t)dt=\int \frac{B(\varepsilon-Blv)d}{Rm}dt \Rightarrow \frac{dx}{dt}=\frac{B\varepsilon...

Homework Statement Homework Equations euler ##e^{ix} = cos(x) + i*sin(x)## ##e^{-ix} = cos(x) - i*sin(x)## The Attempt at a Solution I'm starting with differential equations and I'm trying to understand this solution including complex numbers: First we determine the zeros. I understand that...

Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...

$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$ with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\partial T}{\partial x}\right|_{x=0}=2A\cos^2\left(\frac{\omega t}{2}\right)=A(\cos\omega t+1)$$ where $A=V_0^2/(8RhL)$...

Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...

I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...

Hey all, I hope this is the correct forum section to post this in. I heard about this problem from a youtube video but I've not been able to simulate it because the video was meant only for an introduction into PID control. Here's the problem: A remote control helicopter is hovering just...

Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework EquationsThe Attempt at a Solution substituting ##x## and ##y## I get...

Homework Statement [/B]Homework Equations [/B]The Attempt at a Solution [/B] Could you tell me is there something specific that I need to the with sin(xy)? Thanks

Hi! When we want to look at different singular points for e.g Bessel's eq. $$u´´(x) + \frac{u'(x)}{x} + (1- \frac{n^2}{x^2})u(x)$$. We usually evaluate the equation letting x= 1/z. But I don't algebraically see how such a substitution ends up with $$w´´(z) +( \frac{2}{z}-...

Hey all, I don't understand what makes a differential equation (DE) linear. I found this: "x y' = 1 is non-linear because y' is not multiplied by a constant" but then also this: "x' + (t^2)x = 0 is linear in x". t^2 also isn't a constant. So why is this equation linear?

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081

COPYRIGHT strictly reserved to A. K. Nandakumaran, P. S. Datti & Raju K. George, Department of Mathematics, IISc Bangalore. Duplication PROHIBITED. Lectures: http://www.nptel.ac.in/courses/111108081/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108081