# What is Differential equation: Definition and 1000 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.

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1. ### Difference between resonance in undamped vs damped mass-spring-dashpot

1) "Undamped system is forced at the same frequency as one of its natural frequencies." Consider the 2nd order differential equation $$\ddot{x}+\omega_0^2x=F_0\cos{\omega t}\tag{1}$$ which models a mass attached to a spring (attached to a wall) with spring constant ##k## and...
2. ### Prove that solutions to autonomous first order DE can't have local maxima

Here is one argument. Suppose we have a solution ##y## such that ##y(t_0)=y_0## and ##y'(t_0)=f(y_0)=0##. ##y(t)=y_0## is a solution since ##\dot{y}(t)=0## and so ##\dot{y}=f(y)=f(y_0)=0##. I am aware of uniqueness theorems for linear differential equations. I don't remember seeing such a...
3. ### Series RLC circuit connected to a DC battery

How do I solve the differential equation? Please give me a hint.
4. ### I LIF Neuron Equation Solution for arbitrary time-dependent current (Neural Dynamics)

In the book Neural Dynamics: https://neuronaldynamics.epfl.ch/online/Ch1.S3.html There is a solution to the following differential equation (LIF Neuron) for arbitrary time-dependent current. I was trying to figure out the steps the author took to get to the solution. Original Equation: Solution:
5. ### Help me solve differential equation please involving a fraction and a square root

I'm trying to solve the following differential: ##\frac{\dot x}{\sqrt{y(1+\dot x^2)}} = \text{const}## ##\dot x## is the derivative with respect to ##y##. How do I solve it so that I end up with ##x(y)## solution ? You can find this here, but there're 2 problems: 1) I don't understand what...
6. ### I Differential equation using power series method

I am attempting to solve this differential equation with power series I came with the following solution but I doubt it is correct. Since x=1 we get: I doubt its correctness because it looks messy. Also the convergence radian R goes to 0, giving only a solution for x=0 which is not...
7. ### I Solutions to Simple Harmonic Motion second order differential equation

All simple harmonic motion must satisfy $$\frac{d^2s}{dt^2}=-k^2s$$ for a positive value k. The most well known solution is the sinusoidal one $$s=Acos/sin(\omega t + \delta)$$ A is amplitude, ##\omega##is related to frequency and ##\delta## is phase displacement. My lecturer said that there...
8. ### Apostol Problem on ODE applied to Population Growth

First of all, a few observations 1) It is not clear if the ##t_1## used in problem 14 is the same ##t_1## from problem 13 where ##x(t_1)=\frac{M}{2}##. However, if it were, then the problem seems like it wouldn't make too much sense because we'd have ##M=2x_1## and that'd be it (though this...
9. ### Determine whether ## S[y] ## has a maximum or a minimum

a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##. Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##. Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
10. ### Proofs about the second-order linear differential equation?

Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
11. ### I Question about solving linear first order non-homogeneous ODEs

A general equation for linear first order non-homogeneous ODE is: ## y' + a(x)y = b(x) ##. The procedure to solve ( assuming ## a(x) , b(x) ## are continuous so that the fundamental theorem of calculus could be used ) it is to multiply it by ## e^{A(x)} ## ( here ## A'(x) = a(x) ## ) s.t. ##...
12. ### Solve the given differential equation

My interest is only on the highlighted part, i can clearly see that they made use of chain rule i.e by letting ##u=1+x^2## we shall have ##du=2x dx## from there the integration bit and working to solution is straightforward. I always look at such questions as being 'convenient' questions. Now...
13. ### Solve the given differential equation

I am on differential equations today...refreshing. Ok, this is a pretty easier area to me...just wanted to clarify that the constant may be manipulated i.e dependant on approach. Consider, Ok I have, ##\dfrac{dy}{6y^2}= x dx## on integration, ##-\dfrac{1}{6y} + k = \dfrac{x^2}{2}##...
14. ### A Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution

What is the best way to solve numerically the following equation using Comsol 5.3. ##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}## ##T(0,t)=1## ##T(\infty ,t)=0##...
15. ### Solve these two coupled first-order differential equations and sketch the flow

Hi, unfortunately, I have a problem to solve the following task The equation looks like this: $$\left(\begin{array}{c} \frac{d}{dt} x(t) \\ \frac{d}{dt} y(t) \end{array}\right)=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)$$ Since the following is true ##\frac{d}{dt}...
16. ### Mathematica Problems solving this differential equation for a Pendulum with Mathematica

Hi, unfortunately, I have problems that Mathematica does not solve the differential equation. The task is as follows and it is about the task c In the Mathematica Notebook, the following was written for task c "You can use the following two lines of code to produce the solutions of the...

32. ### Repeated roots of a characteristic equation of third order ODE

The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
33. ### Differential equation of vector field

I was thinking of using the chain rule with dF/dx = 0i + (3xsin(3x) - cos(3x))j and dF/dy = 0i + 0j but dF/dy is still a vector so how can it be inverted to get dy/dF ? what are the other methods to calculate this?
34. ### Solution to Differential Equation with Limit Boundary Condition

The original differential equation is: My solution is below, where C and D are constants. I have verified that it satisfies the original DE. When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
35. ### What does the differential equation answer mean?

Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it. *Arrows are just a continuation of the main formula*

47. ### Solve the first order differential equation

From my working...I am getting, ##xy=####\int x^{-1/2}\ dx## ##y##=##\dfrac {2}{x}##+##\dfrac {k}{x}## ##y##=##\dfrac {2}{x}##+##\dfrac {6}{x}## ##y##=##\dfrac {8}{x}## i hope am getting it right...
48. ### I Problem with integrating the differential equation more than once

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...
49. ### Solve this first order differential equation

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...
50. ### I Definition of order of a partial differential equation

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...