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

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Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
1
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0answers
17 views

what is degree of given PDE…

When the given differential eqn is completely free from radicals the the final exponent on the highest order derivative amounts degree of given differential eqn. In present case it is 3 or 6. i.e. ...
0
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1answer
41 views

Set up differential equation

As people get older, they perceive time differently. The older one is, the faster time goes by. To quantify this issue, we create a model: The entire perceived period of time shall be $w(t)$ . A ...
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2answers
25 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
0
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0answers
15 views

Properties of the solution of an initial value problem

I have an IVP which can not be solved for explicitly of the form: $y''(t) = f(y)(y')^2 +ay'-g(y)$ $y(0)=0, y'(x_1) = h(y)>0$ with $y,y',y'' \in \mathbb{R}_+$, $x \in [0,x_1]$ and I know that the ...
1
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1answer
42 views

Given a solution of a differential equation, determine the differential eqution itself

Sorry if my layout is bad, I'm new. So this question was asked a couple of years ago on an exam about differential-equations. Suppose you have a third order differential-equation with the following ...
3
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2answers
33 views

How to solve differential equation $\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$

My task is to solve for $y$ from: $$\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$$ I have been given the answer, but I would like to calculate this myself also. $\lambda$ is a ...
0
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1answer
31 views

differential equation degree doubt

$dy/dx = sin^{-1} (y)$ this is a form of $dy/dx = f(y).$ so degree should be $1.$ but if i write it as $y = \sin(dy/dx).$ then degree is not defined as it is not a polynomial in $dy/dx $ please ...
0
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1answer
50 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
0
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0answers
14 views

monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
2
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1answer
23 views

the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$

Find all possible $(\lambda,u)$ where $\lambda \in \mathbb R$ and $u\ne0$, to the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$ My Effort: for ...
0
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0answers
17 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
1
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1answer
19 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...
0
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1answer
63 views

Solving a Linear ODE

Solve the following linear ODE $$3t^{2}y'+t^{3}y=cos(t)$$ What i tried Since this is a linear equation, i used the integrating factor method. First i didide both the LHS and RHS of the equation by ...
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0answers
19 views

Example: Solve a Second Order Nonhomogeneous ODE with Constant Coefficients by Variation of Parameters (2R-17)

Problem to solve: $$(D^2-2D+1)y=\frac{e^x}{x^3}$$ Answer in text: $$y=(c_1+c_2x)e^x+\frac12\frac{e^x}{x}$$ Our solution begins by rewriting the ODE in a more familiar form: ...
0
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1answer
32 views

Study of systems of Linear Differential Equations?

Is there any area of mathematics that deals with and formalizes systems of Linear DEs, akin to how Linear Algebra deals with systems of linear equations? Does it provide any insightful results?
0
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1answer
29 views

Linearity of a differential equation

The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one: $a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + ...
11
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0answers
3k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
2
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0answers
21 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
4
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0answers
32 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
0
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1answer
37 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
2
votes
1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
1
vote
1answer
37 views

Why do we need sturm liouville form to solve ODE?

What is the reason that we have to recast a 2nd order ODE into SL form to find its eigenfunctions? for example, let $Ly=y''+y'+\frac{y}{4}=-ky$, boundary conditions $y=0$ at $x=0$ and $y-2y'=0$ at ...
0
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0answers
32 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
0
votes
1answer
29 views

Pass the lower limit to $-\infty$ for an integral of positive function

Hello I have an very elementary calculus problem. Let $\phi(\eta)$ be a real value function satisfying \begin{equation} \phi(-\infty)=1,\quad \phi(+\infty)=0, \end{equation} Let $g$ be a positive ...
0
votes
1answer
48 views

Solving a Variable Separable Differential Equation

The equation is $$y'=\frac{1}{18}x(81-y^2)$$ with $y(0)=81$, and I have to solve for an equation of the form $y(x)$ So I do $$\frac{dy}{(81-y^2)}=\frac{1}{18}x \ dx$$ I integrate both sides, and ...
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0answers
29 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
0
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0answers
16 views

Modulus of Green's function

Consider the nonlinear differential equation $$y'' = f(x,y,y')$$ together with the boundary conditions: $y(\alpha) = A$ and $y(\beta) = B$. Now $y(x)$ is a solution of this problem if and only if ...
0
votes
1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [on hold]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
0
votes
1answer
38 views

Solving an equation by Laplace transform

Consider the following equation: $$ y^{\prime\prime}(x) +x = \int _0 ^x (x-u)y(u)du \qquad y(0)=0 \quad y^{\prime}(0)=1$$ I solved it by Laplace transform and got $-\sinh x$ as a solution. It is ...
2
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1answer
21 views

Extracting differential equations [duplicate]

$$\frac{dx}{dy} = \frac{x(\alpha - \beta y)}{y(\delta x - \gamma)}$$ How do I extract two differential equations (y as a function of x and x as a function of y) from the equation above? I could ...
0
votes
2answers
46 views

Solving a first order ODE

Consider the initial value problem $$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$ (a)Determine how the solution behaves as $t$ tends to infinity. (b)If $y_{0}=2$,find the time $T$ at which the solution ...
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0answers
20 views

Particular Solution of ODE 2?

today I found an interesting example considering viscoelasticity. While I was solving the given ODE I wondered, how the authors came to the solution they gave ... (the thing I am talking about is ...
0
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1answer
21 views

Second order differential equation, physics.

I need your input on this exercise I'm doing: "A 2-kg mass is suspended from a string. The displacement of the spring-mass equilibrium from the spring equilibrium is measured to be 50 cm. If the mass ...
0
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0answers
14 views

Find a Lyapunov function

How can I find a function of Lyapunov ? is there specific methods ? For exemple, how can I find the Lyapunov function of ...
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2answers
36 views

Where is f and gnot differentiable? [on hold]

is my answer correct ???? I try to solve it but not sure if correct or not please help
0
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0answers
14 views

How can I solve this differential equation with fourier series?

Find a formal solution $u(x; y)$ by using Fourier series. (Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with ...
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0answers
4 views

Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
0
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1answer
29 views

Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ [on hold]

Find the Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ satisfying the condition $y(0)=0, y(4)=3.$
0
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1answer
50 views

How to solve $y'''-y=x+1$?

Solve ODE: $$y'''-y=x+1$$ To find the particular solution, I thought to impose $$y_p(x)=ax^3+bx^2+cx+d$$ Fair Enough? Or should I consider other?
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1answer
16 views

Solving first order discrete differential equation

I have a question about solving a first order discrete differential equation. The equation is $x' = Ax$ with $x_{0} = x0$ I found Runge Kutta could solve the differential equation, but required ...
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0answers
23 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
1
vote
1answer
45 views

Prove that there is at most one function that verifies

let $f$ be a function defined on $\mathbb R$ of class $C^2$ and $g$ is a function of class $C^2$ Prove that there is at most one function that verifies $$g(x)=f(x)+\int_{0}^{x} (x-t)g(t) \, ...
0
votes
1answer
13 views

Can a first-order autonomous differential equation have a single steady state at x=0 that is not approached exponentially fast?

I know that x' = kx is exponential growth. I've tried to come up with some solutions. My first solution is x' = -x^3, which has an asymptotically stable point at x=0. It is approached more slowly, ...
1
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4answers
63 views

Solving a homogeneous system of three ODEs with variable coefficients.

I am interested in solving the following system of ODEs: $$ \begin{pmatrix} x'(t) \\ y'(t) \\z'(t) \end{pmatrix} = a \begin{pmatrix} 0 & -B_2 & B_1 \sin \omega t \\ B_2 & 0& -B_1 ...
0
votes
1answer
28 views

How to find general solution of PDE

How to find general solution of equation $$ U_{xy}- \frac{U_x}{y} =0 ? $$ My approach: $$ U_{xy} = \frac{U_x}{y}. $$ Integrate w.r.t $x$ $$ y \ U_y = U + c $$ integrate w.r.t y I don't know how to ...
2
votes
1answer
23 views

Heat Equation, possible solutions

NOTE: This is a homework problem. Please do not solve. I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the ...
2
votes
1answer
33 views

Intuition to solving partial differential equations

I do not understand how to solve PDEs using the geometric method. I just do not understand the logic behind the solution. For example, the constant coefficient equation $$au_x + bu_y = 0,$$ where a ...
2
votes
1answer
43 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
1
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1answer
49 views

What Did I Do Wrong When Solving For This 2nd Order Differential Equation? (answered myself)

$$ \frac{y''}{y'}+y' = f(x) $$ I set the following to be true: $$ y = \sum_{n=0}^{\infty} a_n x^n $$ $$ f(x) = \sum_{n=0}^{\infty} b_nx^n $$ Therefore: $$ y'' = y'(f(x)-y') $$ $$ ...