differential equation Definition and Topics - 143 Discussions
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
(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?
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...
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...
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...
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! 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...
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...
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} +...
I have a problem. The task is to develop an differential equasion of the airflow of a balloon. I know that it is dependent on the volume and pressure. But I can't get a good differential equasion. Can someone help me?
[Thread moved from the technical forums, so no Homework Help Template is shown]
I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck:
$$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
How does one solve the partial differential equation Uxx+Uyy+Uzz=C when C is non-zero. Here U is a function of x,y and z where (x,y,z) lies in the ball centered at 0 of radius 1 and U=0 on the boundary. Uxx, Uyy and Uzz denote second partial derivatives with respect to x, y and z.
Any hints on...
From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method.
For example, 2xy''+y'+y=0
after going through all the math i get
y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...]
y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...]
I know those are right...
If we take a flat universe dominated by radiation, the scale factor is ##a(t)=t^{1/2}##
which can be derived from the first Friedmann Equation:$$(\dot a/a)^2 = \frac{8\pi G}{3c^2}\varepsilon(t)-\frac{kc^2}{R_0^2 a(t)^2}$$
But suppose I want to show this using the second Friedmann Equation
(Also...
When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE
\psi''-y^2\psi=0
solved in the limit where y tends to infinity.
I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...
The ODE is:
\begin{equation}
(y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0
\end{equation}
Where y(x) and z(x) are real unknown functions of x, m is a constant.
I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
Homework Statement
Given that ##x=\phi (t)##, ##y=\psi(t)## is a solution to the autonomous system ##\frac{dx}{dt}=F(x,y)##, ##\frac{dy}{dt}=G(x,y)## for ##\alpha < t < \beta##, show that
##x=\Phi(t)=\phi(t-s)##, ##y=\Psi(t)=\psi(t-s)##
is a solution for ##\alpha+s<t<\beta+s## for any real...
Homework Statement
Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height.
Homework Equations
I will include some pictures to show what I'm doing
The Attempt at a...
Homework Statement
I have to calculate the critical points of the following system.
$$x'=cx+10x^2$$
$$y'=x-2y$$
The Attempt at a Solution
So I solve the system
$$cx+10x^2=0$$
$$x-2y=0$$
So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
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 Equations
The Attempt at a Solution
substituting ##x## and ##y## I get...
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).
I've also found a recursion relation for the equation to be:
An+2 = -An / (n+2)(n+1)
I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?
Any help would be...
Hey, this is how i tried solving the differential equation
The solution however does not match the general solution of the equation. Also differentiating it twice does not give me the previous equation. Please tell me if i did some mistake while solving.
I already know how to solve by finding...
Homework Statement
The question I am working on is the one in the file attached.
Homework Equations
y = u1y1 + u2y2 :
u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)
The Attempt at a Solution
I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
I've been having a sign problem while deriving the permittivity formula using Drude model,
and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated...
Hi all,
I'd appreciate help in calculating the voltages in the circuit shown. I thought it should be fairly straight forward, but it has me stumped. This is a sample-and-hold circuit for an ADC -- the switch is closed to charge the hold capacitor with the sample voltage and then opened to...
Hello all, I want to say thank you in advance for any and all advice on my question. My classical mechanics textbook (Marion Thornton) has been taking me through motion for a particle with retarding forces.
The example it keeps giving is:
m dv/dt = -kmv
which can be solved for:
v = v0e-kt...
I am aware that hypergeometric type differential equations of the type:
can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...
I have a system with three inductors connected together at a common point.
The unconnected ends of each inductor is connected to an independent voltage source.
Basically I want to get three expressions for the dynamics of the currents with V1, V2 and V3 as inputs.
i.e. i need to eliminate the...