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

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1answer
17 views

Struggling to solve differential equation once integrated

The question is this: A body falling vertically though the air has speed $v$ at time $t$, related by the differential equation $\frac{\delta v}{\delta t} = g - cv$ where g and c are positive ...
0
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1answer
15 views

General Methods to Solve First-Order PDE

Question is as simple as: What are the different methods for solving a first-order PDE? I'm aware of nearly all forms of Method of Characteristics - Lagrange Method, Charpit's Method. I'm ...
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0answers
17 views

Differential Equation-Cannot find the method to this types of problem.

This is the problem I don't know the exact way to do this type of problem,can anyone kindly give me.
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0answers
7 views

Solving a non-linear ode system

I'm having difficulties to solve a problem, everything that I try leads to a harder problem, so if there are any tips it would help a lot. This is the problem: Find all non-trivial solutions ...
1
vote
1answer
62 views
+50

Model for spread of infection, with vaccination

I'm trying to solve following problem: $N = 10^6$ ... number of people $ir = 8\% $ ... infection rate time unit - 1 day And when there are 3% of population infected, vaccination begins. Its effect ...
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2answers
25 views

Differential Equation $y'=\sqrt[3]{(4x-y+1)^2}$

I'm having problem with sovling this equation. $$y'=\sqrt[3]{(4x-y+1)^2}$$ I know I have to use change of variables e.g. $z=4x-y+1$ and $z′=4-y'$ but then I am not getting anywhere.
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0answers
8 views

Determination of values for $\alpha$ so the system becomes global asymptotic stable.

The following system given below: and $\alpha$ is a real number. Since the system is linear I look at the determinant and the trace of this system. The matrix for the system is: $$ A ...
3
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0answers
41 views
+50

Counter example for uniqueness of second order differential equation

I have a second order differential equation, \begin{eqnarray} \frac{d^2 y}{d x^2} = H(x) y \quad \quad \quad * \end{eqnarray} where, $H(x) = \frac{sech(x) sech(x)}{x + \ln(2 \cosh(x))}$ . Plot of ...
-1
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1answer
26 views

Solving differential equation $y'(x)=e^{a(x+y)}+3e^{ay}$ if $a \neq 0$

The differential equation is given below marked with (*). I have to determine the complete solution to (*) if $\alpha \neq 0$. I'm not sure how to approach this problem. I am thinking that it has ...
0
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1answer
29 views

Proving that a function maintain a certain equation

Hey guys so here a new question, I need to prove that the function $$g(x,y,z)=f(\frac{1}{y}-\frac{1}{x},xye^{\frac{-z^2}{2}})$$ maintain the equation $$x^2g_x+y^2g_y=-\frac{x+y}{z}g_z$$ while ...
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2answers
28 views

Solving Differential Equation $y'=\sin(y-x-1)$

I'm stucked with sovling this equation. $$y'=\sin(y-x-1)$$ I know I have to use change of variables e.g. $z=y-x-1$ and $z'=y'-1$ but then I am not getting anywhere
-3
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2answers
47 views

Differential Equation $y'-1=\frac{y}{x}-\frac{y^2}{x^2}$ [on hold]

Can anyone help me solve this equation ?? I think it's Riccati equation but no answer is given in the question $$y'-1=\frac{y}{x}-\frac{y^2}{x^2}$$ The answer must contain $y$ and $x$ and free of ...
0
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1answer
14 views

Understanding shift to polar coordinates in the newtonian central force system of ODE's

This is from Hirsch, Smale and Devaney chapter 13. The larger context is moving towards blowing up the singularity at the origin of the system. The second order ODE is defined, $X:t\rightarrow ...
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1answer
41 views

how to solve this differential equation $(3(x^5)+3(x^2)(y^2))dx + (2(y^3)-2(x^3)y)dy = 0$

This is my first question here. I tried to solve this ODE. This is the Wolfram's answer but there's a step-by-step solution. :( Thanks
2
votes
1answer
16 views

Solving a separable integral equation: $y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$

Solving integral equation. My answer is wrong. Where do I make a mistake? $$y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt $$ $$ y'(x) = \frac{d}{dx} \int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$$ $$ ...
1
vote
1answer
24 views

Differential equation corresponding to a linear system of differential equation.

Consider linear system of differential equations $$\frac{dx}{dt}=ax+by$$ $$\frac{dy}{dt}=cx+dy$$ my question is how to find the second order linear differential equation corresponding to above ...
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0answers
18 views

Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
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0answers
45 views

How do I solve for simple pendulum?

For example, I have the Euler-Lagrange equation for a simple pendulum: \begin{eqnarray*} \frac{d^2\theta}{dt^2}+\frac{g}{l}\theta &=& 0 \end{eqnarray*} How do I find $\theta$? I know that ...
0
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0answers
24 views

Linearising a system of ODE's

So I have this ODE: $$x''=\frac{-27}{13}x'+\frac{35}{13x}-\frac{4843}{283}$$ The question asks me to write a MATLAB script to linearise this ODE around any specified x value, as far as I know only ...
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0answers
18 views

Implication of nonzero radius of convergence

I'm reading the proof of the theorem that the equation $u'' + p(z)u' + q(z)u = 0$ where $p,q$ have poles of order one and two respectively, has a solution of the form $u(z) = z^a \sum_{j=0}^{\infty} ...
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votes
1answer
26 views

A nice question related to method of characteristics

Let $ \alpha$ be real number and $h=h(x)$ be a continuous function in $\mathbb{R}$.Consider following initial value problem: $$yu_x + xu_y=\alpha u, u(x, 0) =h(x) $$ Then a) Find all points on ...
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0answers
24 views

Differentitaion of a non-linear equation using FDM method

I'm a Ph.D student of Hydraulic structures. I'm reading a paper in that the equation $(II)$ below is obtained by differentiating the equation $(I)$ using FDE (Finite Difference Equation) method and ...
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2answers
33 views

Differential Equation $y^2\frac{dy}{dx}+2xy=e^y$

I am having a problem with solving this equation. I've tried different ways but nothing works. $$y^2\frac{dy}{dx}+2xy=e^y$$
2
votes
1answer
47 views
+100

Positiveness of energy of differential equation

Let $x(t) : [0,T] \rightarrow \mathbb{R}^n$ be a solution of a differential equation $$ \frac{d}{dt} x(t) = f(x(t),t). $$ In addition we have functions $E :\mathbb{R}^n \rightarrow \mathbb{R}$ ...
0
votes
1answer
28 views

Solution of ODE: (f'(x))^2 - (f(x))^2+ 1 = 0

A few days ago, I came upon the discovery that the arc length of $ f(x) = cosh(x) $ is equal to the area between any two points of its points. It made me curious to find other functions that satisfy ...
3
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1answer
44 views

4th order differential equation from Euler-Lagrange

I am trying to extremise the functional $\int{[y + \frac{1}{2}y^2 - \frac{1}{2}(y^{''})^2]}dy$ and so from Euler-Lagrange I get the differential equation $1 + y + y^{(4)} = 0$ and I have no idea how ...
4
votes
1answer
2k views

Solution verification: Find the orthogonal trajectories of the family of curves for $x^2 + 2y^2 = k^2$

I need help with the following question: Find the orthogonal trajectories of the family of curves for $x^2 + 2y^2 = k^2$ I have taken the following steps, are they correct? From what I ...
3
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2answers
25 views

Relation between differential equations and sequence recursions

It's obvious that there is a strong relation between linear recursions of sequences and linear differential equations. The common methods for solving them are nearly identical. For example, the ...
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0answers
15 views

Fundamental system of solutions and Green`s function

Suppose we have the fundamental system of solutions of equation $$y''(x)+(u_0x+u_1)y(x)=f(x)$$ with initial conditions $$y(0)=y_0,~ y'(0)=y_1.$$ Its fundamental system of solutions is ${\rm ...
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1answer
377 views

Green Function First Order DE

I'm having a bit of trouble working out the Green function for $$y' = f(x)$$ when the boundary condition is something like $y(a) = y_0 \neq 0$. I'll do it as though $y(a) = 0$ and at the end modify to ...
2
votes
1answer
24 views

Show that all the solutions of the given differential equation are bounded

Let $ f:[0,\infty)\rightarrow \Bbb R$ be a bounded and continuous function. Show that every solution of the differential equation $$y''+2y'+5y=f(t,)\quad t\ge 0$$ is bounded on $[0,\infty)$. By using ...
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1answer
25 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rcosϴ, r = \frac{x}{cosϴ} , \frac{dr}{dx} = \frac{1}{cosϴ} $$ $$\frac{dA(x)}{dx} = ...
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3answers
25 views

Third Order Differential Equations

I am having trouble solving the third order differential equation $y'''+y'=0$ It was given to me in a quiz (which I got wrong) with boundary conditions $y(0) = 0$ $y'(0)=2$ $y(\pi)=6$ I know that ...
0
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1answer
26 views

Converting higher-order ODE to first order ODE

given $y''' + 2y'' -5y' = 2y + 5y^3$ convert to a system of first order equations. My question is do we need to make substitutions for $y$ and $y^3$ or are we only concerned with the derivatives, if ...
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0answers
8 views

On what interval does the following ODE have a solution

The ODE is $y'=\frac{x}{y}$ with the initial condition $y(0)=e^{\frac{i\pi}{4}}$. As solution I get $\phi(x)=\sqrt{x^2+e^{\frac{i\pi}{2}}}$ which in turn gives ...
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0answers
18 views

How do I determine values of b, so the difference equation (*) becomes global asymptotic stable?

The difference equation in the picture down below is marked with (*): I don't know where to start. In which direction should I be looking at? I tried to start looking at the characteristic ...
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2answers
36 views

Solving $xy'+y=x^{k}$

Find a solution to: $$xy'+y=x^{k}$$ Where $k>0$ and $f$ and $f'$ exist. I understand that we can take the Laplace of all of the terms and then find the inverse Laplace transform to get a ...
0
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1answer
415 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 ...
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0answers
15 views

ODE, Stability Analysis, Graduate level [on hold]

What are good resources online to work on problems for Graduate level candidacy exam on ODE. We used Grimshaw's and Perkov's book. If theres' any solution manuals do let me know. Appreciate any ...
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0answers
18 views

Analytic answer of a differential eq. [on hold]

I want to solve a diff eq analytically. y' = $x^2$ + y But i don't remember technics. Please someone help me. Answer in Matlab : y = constant * $e^x$ + 2x - $x^2$ - 2
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1answer
79 views
+50

An MCQ on Greens function

$$G(x,t) =\begin{cases} a+ b\log t & \text{if $0<x<t$ } \\[2ex] c+ d\log t & \text{if $t<x<1$ } \end{cases}$$ is a Greens function for $xy''+y'=0$ subject to $y$ being bounded as ...
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0answers
22 views

System of Ordinary differential equations.

Consider the system $$\frac{dx}{dt}=(1+x^{2})y$$ $$\frac{dy}{dt}=-(1+x^{2})x,t\in\mathbb{R.}$$ With initial condition $(x(0),y(0))=(a,b)$ ,Then the system has solution $1.$ Only if $(a,b)=(0,0)$ ...
0
votes
1answer
13 views

Solving solution given initial condition condition

Suppose we know that: $$u_t=ku_{xx},~~~~~~~~0<x<l,~~~t>0$$ and $$u(x,t)=\sum_{i=0}^\infty[C_n~cos(n\pi x/l) ~e^{-w_nkt}]$$ where $w_n=\frac{n\pi}{l} ~~~ for~~n=1,2,3,...$ What if the ...
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0answers
19 views

For a nondecreasing map, if $\xi(a) < \eta(a)$, then $\xi(t) < \eta(t)$ for all $t \in [a,b]$.

I am studying the following theorem from Morris Hirsch's second paper on systems of differential equations which are competitive or cooperative: Let $V \subset \mathbb{R}^n$ be on open set and ...
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0answers
25 views

Error of the Numerov Method

The Numerov method is an iterative algorithm for solving second order differential equations. A full derivation is here on the Wikipedia page: https://en.wikipedia.org/wiki/Numerov's_method. I am ...
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1answer
23 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.
2
votes
1answer
19 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
0
votes
2answers
33 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
0
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0answers
20 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...