Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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The Picard-Lindelof theorem on Wikipedia

On the Wikipedia entry of Picard-Lindelof theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval. There is a lemma used in this ...
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35 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
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1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
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3answers
168 views

How do I solve $y' = \sin(x - y)$?

How can I solve the differential equation: $$y'=\sin(x-y)$$ Could I do this? $$\frac{dy}{dx}= \sin x \cos y - \sin y \cos x$$ But, how would I continue?
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harmonic oscillator exercise

An object with a mass of 8 kg stretches a spring over 0.06 m This object is drawn further down to 0.30 m and set in motion by an upwardly directed velocity of 0.30m/s in a substance that has a ...
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1answer
44 views

Nonlinear 1st order ODE

$$y'(x)=\frac{\cos (y(x))+y(x) \cos (x)}{x \sin (y(x))-\sin (x)}$$ Did I enter LaTeX correctly? I am self-learning the differential equation from a textbook and I need some help with above equation. ...
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2answers
146 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
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1answer
22 views

Conditions where Euler's method over-estimates or under-estimates

Is there a rule that characterizes when Euler's method over-estimates or under-estimates? For example, "if $f(x)$ is increasing, then Euler's method underestimates," or something similar?
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1answer
24 views

In what cases are the eigenvalue equal to the pole points?

I have a transfer function in form of a matrix and want to determine the stability of the whole system. Now I'm wondering if I need to calculate the pole points or the eigenvalue. A friend of mine ...
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3answers
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what can we say about the solution of the equation $y'=-y^2$ just by looking at it. [without finding its solution]

I trying to understand differential equations without finding their solution. This is a simple one, so I can verify the ideas from the solution. All ideas are appreciated, since they will help my ...
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1answer
39 views

How to solve this differential equation sinusoidal?

I can't find how to separate variables. $$y= \sin(xy')$$
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3answers
45 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
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1answer
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Finding first few terms in power series expansion of general solution [closed]

I need to find the first four nonzero terms in the power series expansion for the general solution to the differential equation $y''-x^2y=0$. My work thus far: ...
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1answer
31 views

Study the stability of the following system

Study the stability of the system $x'=Ax+b(t)$ where $$A=\left(\begin{array}{rcl} c & 0 & 0\\ 0 & 1 & 5\\ 0 & -5 & 1 \end{array} \right)$$ Is there an easy way to solve this ...
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1answer
19 views

Differentiation - Equilibria

The size z(t) of a hailstone evolves according to the differential equation $ \frac{dz}{dt}= A\sqrt(z) - B\sqrt(z^3) $ where A and B are positive constants. Without solving the differential ...
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Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
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numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
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Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
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2answers
49 views

Proving an equation has a unique solution

Prove that $ a^2-2ax+(2ax-a^2N)e^{(-a/x)}=0$ has a unique solution. (Note: $x>0, a>0, N$:positive integer). Thanks.
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How to solve the following an ODE?

Let $x,y,z$ be a given point in $\mathbb{R}^3$. How to solve $(x'(t),y'(t),z'(t))=(x(x+y+z), y(x+y+z),z(x+y+z))$?
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0answers
79 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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1answer
19 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
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1answer
22 views

Study the stablility of this ODE

I have to study the stability of: (a) Stability of $x''+kx'+(2k-1)x=0$. (b) Asymtotic stability of $t^2x''+tx'+x=0$. How can i solve this problems? Please any help, to understantd this problems.
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Prove that the system is asymptotically stable.

Let $A_0,\cdots,A_m$ constant matrices. Suppose that the eigenvalues of $A_0$ have negative real part. Prove that $$y'=(A_0t^m+\cdots+A_m)y$$ is asymptotically stable. Hint: ...
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1answer
64 views

Consider the following differential equation$ y'' + 5y' + 4y = 0$.

a) Determine a system of equations $x' = Ax$ that is equivalent to the differential equation. b) Suppose that $y_1, y_2$ form a fundamental set of solutions for the differential equation, and $x(1), ...
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Runge-Kutta for newton's law with dependency

I'm trying to determine the changes (position and velocity) on a mechanical system during a step of time. I have a mobile mass whose position (everything is 1D-only) is denoted $x(t)$, velocity $v(t)$ ...
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21 views

the assumption in variation of parameters in solving ODE

according to (Zill and cullen)_page53_Advanced_engineering_mathematics : in first order linear ODE when we use the technique of variation of parameters to find the solution of the inhomogeneous ...
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1answer
78 views

Differential Equation Word Problem

I'm new to this site. I'm taking a differential equations course this summer and this question is from my first homework assignment. I was able to do all of my homework questions that just required me ...
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0answers
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Initial value problem with intermediate value

The Picard Lindelöf theorem I know always assumes that we specify the value at the left end of the time-interval Picard Lindelöf. Is it true that $x'(t) = f(x(t))$ has a unique solution, in an open ...
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0answers
32 views

Find a solution to a linear system using the D operator and method of elimination

This is from Introduction to Differential Equations and its applications by Farlow. Info: The $D$ operator Let $D$ denote differentiation with respect to $t$, and let $D^2$ denote the second ...
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4answers
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$y' = \exp(-\frac{y}{x}) + \frac{y}{x}$ [duplicate]

Could you help me to solve this differential equation: $$y' = \exp\left(-\frac{y}{x}\right) + \frac{y}{x}$$
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I need help figuring this error percentage homework problem.

Question: Government economists in a certain country have determined that the demand equation for soybeans is given by $p = f(x) = \frac{55}{2x^2 + 1}$ where the unit price p is expressed in dollars ...
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4answers
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I need help figuring this error percentage homework problem.

Question: The period of a simple pendulum is given by $T=2\pi\sqrt{\frac{L}{g}}$ where $L$ is the length of the pendulum in feet, $g$ is the constant of acceleration due to gravity, and $T$ is ...
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Using matlab ode23s to solve the 6 nonlinear differential equations:

Using matlab ode23s solver i am trying to solve the 6 nonlinear differential equations: Non-linear differential equations: $ \frac{dc_0}{dt}= c_0*(- K_F - K_D - K_N * s_0 - K_P*(1-q_0))$ ...
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Taking the derivative of a summation

I need to know how to solve this, I would prefer a step by step process, and not just a solution please, working with perceptrons in a neural net. n is the number of nodes in a layer, every node is ...
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derivative or differentiation with respect to a sum

I have the function $F(z',z,x,y)$, where $z=z(x,y)$ and $z'$ is the differential of $z$ with respect to its argument, and $x, y$ are the two independent varaibles here. So, $z$ and $z'$ are dependent ...
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differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
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1answer
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What smooth functions are solutions of an autonomous ODE?

Let $y$ be a smooth function, say $y : \mathbb{R} \rightarrow \mathbb{R}$. When can we find a continuous map $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $y'=f(y)$ ? Obviously it's not always ...
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1answer
38 views

Differential Equation Examples for different type of critical point

For a linear system $X'=AX$, there are only limited types of critical points according to the eigen values of $A$. When I want to considering non-linear dynamical system in $\mathbb{R}^2$ and ...
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1answer
24 views

Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...
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The phase portrait of a second order of nonlinear system using matlab

I have the following system $$ \ddot{x} + 0.6\dot{x} + 3x + x^{2} = 0 $$ In the book I'm reading, the phase portrait of the nonlinear system for the aforementioned equation is I would like to ...
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Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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1answer
25 views

Bessel's Equation

How can I solve this Bessel's Equation? $$x^2*y''+x*y'+(x^2-v^2)y=0$$ First I did that: $$y''+\frac{1}{x}y'+\frac{(x^2-v^2)}{x^2}y = 0$$ Then, ...
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Solving a ODE using series

I have to prove that the series $$y(x)=\sum_{n=0}^{+\infty}\frac{x^{n}}{(n!)^{2}}$$ satisfies the ODE $$xy''+y'-y=0$$ When I derivate and substitute in the equation, I get ...
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1answer
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Re-expressing the Schrodinger Equation as a first order expansion.

I am reading an online text on quantum computing and the author expands and re-expresses the Schrodinger equation. I am not really sure as to the intermediate steps he used or what happened to the ...
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1answer
24 views

Definition of “epidemic” when using SIR models

I haven't studied differential equations for a long time, but I have just started looking at material on the SIR model of epidemics. My problem is that the resources that I've looked at haven't given ...
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1answer
26 views

Second Order Linear Equations

I have $y''+y'-2y=0$, $t_0 = 0$ I need to use Abel's theorem So I get $W(y1,y2) = ce^{(-t)}$ What should I do next?
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2answers
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The volume is preserved by the flow: where is the absolute value?

Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics": In changing the variables in the integral, I don't understand why ...
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1answer
27 views

Expressions and equations [closed]

The Murphy's love to have parties. Last Friday, they gave a party and the doorbell rang 15 times. At the first ring, one guest arrived. Each time the doorbell rang after that, two more guests arrived ...
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ODE particular solution (physics)

I have to do this exercise: ($Z(t)=I(t)$, it's printed wrong). I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega ...