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

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1answer
23 views

Finding the Jacobian of a system of 1st-order ODEs

I am trying to find the Jacobian matrix of the following system of 1st-order ODEs. My system is: $\dfrac{dx}{dt} = \left(x-3\right)\!\left(y+x\right) \\ \dfrac{dy}{dt} = ...
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1answer
21 views

Is the linear combination of two solutions of a nonhomogeneous differential equation also a solution

The question reads, if y1 and y2 are solutions of: $y''+x^2y'-e^xy=1$ then is any linear combination of y1, y2 also a solution. I know for a fact that the above statement is true for homogeneous ...
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1answer
46 views

Proving a differential inequality without performing iteration

I'm seeking a better proof of the following fact: If $g$ is a non-negative bounded function, $g(0)=0$ and $g'(t)\leq \sqrt{g(t)}$ for all $t>0$, then $g(t)\leq t^2/4$. The upper bound $t^2/4$ is ...
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0answers
16 views

Is there a way to delineate the parameter of highest influence in a system of differential equations?

So I have a system of nonlinear ordinary differential equations dependent on parameters. These equations can traditionally be solved numerically with robust methods and the solution is well defined. ...
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2answers
26 views

Complex Eigenvalues and systems of 1st-order ODEs

I am trying to solve $\begin{aligned} \dfrac{dx}{dt} &= x+ y \\ \dfrac{dy}{dt} &= -10 x- y \end{aligned}$. My strategy is to let $P = \left[\begin{array}{cc} 1 &1\cr -10 & -1 ...
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2answers
27 views

Parametric integral question

I haven't done something like this in a long time. How do I set something like this up? Can someone help me with the beginning or give me some direction?
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1answer
27 views

Expressing associated Legendre polynomials in terms of unassociated Legendre polynomials

The associate Legendre equation is given as: $(1-x^2)\frac{d^2}{dx^2}y-2x\frac{d}{dx}y+\left[n(n+1)-\frac{m^2}{1-x^2}\right] y=0$ This becomes the standard unassociated Legendre equation for $m=0$. ...
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0answers
24 views

Solving a 2nd-order ODE via Eigenvectors/values

I am trying to solve $x'' - 8 x'+ 25 x = 0$. I found the general solution as $Ae^{4t}cos(3t) + Be^{4t}sin(3t)$, where $A,B$ are arbitrary constants. However, when I write the differential equation as ...
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2answers
139 views

Solving a differential equation by using Laplace transform

I need to solve this equations by using laplace-transform. I tried to solve it but when I reach to the point that it's needed to use partial fraction expansion in order to transform the laplace ...
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0answers
14 views

A 2-variable function with parts of its derivative

Let $f$ be a function with partial continuous derivatives, The derivative of $f$ in $ A=(2,2) $ in the direction from $A$ to $B=(2,5)$ is $6$ The derivative of $f$ in $ A=(2,2) $ in the direction ...
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0answers
16 views

Scalar product with $f(x,y)$ when $ \dot{X} = f(X)$ has periodic orbit

Let $g(x,y), f(x,y) \in C^1: R^2 \to R^2$ such that $f(x,y) \cdot g(x,y)=0$, $\forall (x,y).$ Prove that if $\dot X = f(X)$ has a periodic orbit, then $g$ have a root Intuitively I can see that the ...
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0answers
18 views

Given a 2-variable function, prove that: $xz * z'_x-yz*z'_y = -\frac{1}{2}$

I'm kinda new to this material and was having a hard time solving this exercise, can anyone please show me the correct way of solving this? Let $z(x,y)$ be an equation that satisfies: $ ...
2
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1answer
24 views

Analytic geometry line segments

This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! ...
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2answers
56 views

A system of 1st-oder ODEs

I am trying to solve $\begin{aligned} \dfrac{dx}{dt} &= -2 x - 6 y \\ \dfrac{dy}{dt} &= x - 7 y \end{aligned}$. Let $P = \begin{bmatrix}-2&-6\\1&-7\end{bmatrix}$. My approach is to ...
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1answer
18 views

Solve a particular differential equation

I am quite new to differential equations and I have to solve the following $a(t)+b(t)C(t)+s a(t)C'(t)=0$, where $s$ is some constant. I read about differential equations and at this moment my main ...
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0answers
18 views

ODE and PDE : Solving DE with constant coefficients using differential operators

I want to study the method of finding solution to Differential Equations with constant coefficients using method of operators (finding the particular integral to the equation using inverse operator). ...
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0answers
43 views

Using a Fourier Series to Solve Differential Equation

The problem states to use the fourier series of the function f(t) defined as follows: $f(t)= t+1 , -1<t<0 $ $f(t)=1-t , 0<t<1$ to solve the differential equation: x''+4x=f(t), x(0)=1, ...
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2answers
49 views

Differential Equations and Eigenvectors/Eigenvalues

I am trying to solve $\dfrac{d\mathbf{x}}{dt} = \left[\begin{array}{cc} -4 &1\cr -6 &1 \end{array}\right] \mathbf{x}$ and I need to find the general solution of the system in the form ...
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1answer
27 views

Taylor's inequality question

Totally don't know how to go about doing this. Any help/insight would be appreciated.
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0answers
26 views

this function is known ?: $h_{n+1}(x)=h_n(x)^2+h′_n(x)$

Let $f(x)=x^{1/x}$, so the first derivative of $f(x)$ is $f′(x)=f(x)∗(1−ln(x))/x^2$, and in general, $f^n(x)=f(x)∗h_n(x)$, where $f^n(x)$ is the nth derivative of $f(x)$. I was trying find this ...
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2answers
51 views

Elegant integration method I'm missing…

Compute $r'(t)$, where $$r(t) = \left(\int_1^{t^2} s^6 e^s ds \right) \textbf i + \exp\left(-\int_1^t\arctan(s) ds\right) \textbf j$$ Hint: for the $i$ part, there is a significantly ...
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1answer
39 views

Analyzing the singularity of ODE system

It is asked to analyze the singularities of the system $$\dot{x} = y e^y$$ $$\dot{y} = 1-x^2$$ I've found that the singularities are (1,0) and (-1,0) The linearization of the sysyem give the matrix ...
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0answers
32 views

Initial value problem - How can we find the coefficients $c_j$?

We have the initial value problem $$u'(t)=Au(t) \ \ , \ \ 0 \leq t \leq T \\ u(0)=u^0 \\ u \in \mathbb{R}^m$$ A is a $m \times m$ matrix The eigenvalues of $A$ are $\lambda_j$ and the corresponding ...
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0answers
37 views

Solving higher order pdes

So here's my problem, while you solve the Euler Bernoulli beam Equation by separation of variables, how do I have to prove the separated function of space are orthogonal? If so, are hyperbolic sines ...
2
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1answer
31 views

Order of accuracy of Runge-Kutta method

The following Runge-Kutta method is given by the Butcher tableau: $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =1 & a_{21} = \frac{1}{2} & a_{22} = ...
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0answers
31 views

Time period of oscillations of a point about the function's minimum value?

How am I to go about the following problem? Please do not explicitly solve it. Let $E_0$ be the value of the potential function at the minimum point $\xi$. Find the time period $T_0=\lim_{E\to ...
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0answers
17 views

How to turn the integro-differential equation into an ODE

I want to get the numerical solution of the integro-differential equation by Mathematica but failed. Maybe the first step should be turning that into an ODE, is there some method? {0.01+10 (0.01 ...
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3answers
52 views

Differential equation $y'' \cdot y^3 = 1$

I use these substitutions $y'=p(y)$ and $y'' = p' \cdot p$ to solve the equation, thus I have the consequence of the solution's steps: $$ p'py^3 = 1 \implies p'p = \frac{1}{y^3} \implies \frac ...
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2answers
30 views

Why exponential is ignored in particular solution for impulse response ??

For a system govern by the equation: $$ 2y'(t) +4y(t) =3x(t) $$ To calculate it's impulse response we replace $y(t)$ with $h(t)$ and $x(t)$ with $\delta(t)$ and get $2h'(t)+4h(t)=3\delta(t)$ which's ...
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0answers
9 views

Complex 2 variable differential equation of order 4

Is there any way to obtain all solutions od the following equation: $$ u^{IV}+A\ddot{u}+B\ddot{u}''+i\dot{u}''=0,$$ where $u=u(z,t)$, $z \in \mathbb{C}, t >0$, the dot represents a derivative with ...
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0answers
109 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a ...
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0answers
19 views

How to get this numerical solution of a integro-differential equation

Previously, I ask an NDSolve questions(How to solve the differential equation with Duhamel's integral?), but now a more sophisticated one: ...
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0answers
10 views

Nonuniform partition - euler method

Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, ...
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1answer
19 views

Differential spherical wave equation, why is the result the same for real and imaginary parts?

I wasn't very sure whether to ask this in the physics forum or here, but the question regards mathematics much more than it does physics. The following wave function is given (spherical wave): ...
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1answer
39 views

Prove an inequality for a specific Ordinary Differential Equation

Let $a \in \mathbb R$ Consider the differential equation $$\frac{d^9y}{dt^9}-\dfrac{dy}{dt}+ay=0 \tag 1$$ suppose $\varphi : \mathbb R \rightarrow \mathbb R$ is a solution of $(1)$ on $\mathbb R$. ...
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0answers
6 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
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0answers
6 views

Simple inverse laplace using partial frac not so simple?

When evaluating the step response of a circuit, the resulting Laplace representation is: $\frac{I_{pd}}{s^2 C1 R1}$ If I look this up on a table of Laplace Transforms, this results in ...
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0answers
19 views

Solve differential equation using Liouville formula.

I have $y_1=\left[\begin{array}{rr}e^t\\-e^t\end{array}\right]$ $y'=\left[\begin{array}{rr}2-t & 1-t\\t-2 & t-1\end{array}\right]$. Using Liouville formula ...
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1answer
15 views

Hamiltonian Constant on integral curves

Let $H \in C^{2}(\mathbb{R}^2)$ and let $(x(t),y(t))$ be a solution to the equations $$\frac{dx}{dt} = \frac{\partial}{\partial y} H(x(t),y(t))$$ $$\frac{dy}{dt} = -\frac{\partial}{\partial x} ...
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0answers
12 views

Absolute stability Euler method

I am looking at the following exercise. We suppose that the explicit Euler method is applied at the differential equation of second order $\left\{\begin{matrix} x''(t)+(\lambda+1)x'(t)+ \lambda ...
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3answers
175 views

Is this ODE separable?

I'm preparing for the final in my ODE course by reviewing some past exams and I found this problem. Solve the following equation by the separation of variables method. ...
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2answers
38 views

2nd order Ordinary differential equation, $ay'' + by' + cy = 0$, where $b$ is complex

In particular my question was to solve this equation differentiated with respect to variable t: $$\begin{cases}y'' + i K y' + \frac{M^2}{4} y = 0\\ y(0) = 0\end{cases}$$ After solving with $e^{r ...
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2answers
23 views

An example of a BVP for a second order ODE: $y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$)

I'm looking for an explicit example of a BVP for a second order ODE: $y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$). If you also have the exact ...
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2answers
38 views

Derivatives and solving a system of linear ODEs

I am working on finding the general solution of the system in the form $x=c_1x_1+c_2x_2$ given $\dfrac{d\mathbf{x}}{dt} = \left[\begin{array}{cc} -4 &1\cr -6 &1 \end{array}\right] \mathbf{x}$. ...
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2answers
56 views

The general solution to the differential equation

I am having problems in classifying the differential equation $y''=y(x^2)$ in categories like homogeneous, exact, bernoulli, separable and non-exact so I could have the general solution. Or would ...
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1answer
14 views

Taylor expansion of $f$ in stability analysis of 2-step Adams-Bashforth method

Given the two-step Adams-Bashforth method $$ u_{n+1} = u_n + \tfrac{h}{2}(3f_n - f_{n-1}) $$ find its order. Some notation: $t_n = t_0 + nh$ is the $n$-th node and $y_n = y(t_n)$; $f_n$ ...
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0answers
22 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
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1answer
32 views

find a function $F:[0,2]\to\mathbb{R}$ with continuous derivative in $[0,2]$ and satisfies the following problems of initial conditions [on hold]

find a function $F:[0,2]\to\mathbb{R}$ with continuous derivative in $[0,2]$ and satisfies the following problems of initial conditions: $y''-4y=0$ in $[0,1]$ with $y(0)=0$, $y'(0)=1$ $y''-2y=0$ in ...
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0answers
32 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
0
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1answer
26 views

Solve for $y_1, y_2, y_3$ using Picards' method of Iteration: $y'=1+y^2$ ; $y'(0)=0$

I need to find $y_1, y_2,$ and $y_3$ using Picard's method. What I don't understand is why they initially give $y'(0)$ instead of $y(0)$. $y(0)$ is needed in order to use the method because it goes ...