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

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2answers
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A strange 3rd order ODE

This is the original ODE: $ y^{1/2}y'''+e^{-x}(y'')^{2+c}-(\frac{xy}{x+1})y'=x $ with c is a positive number. $y(0)=1,y'(0)=0,y''(0)=1$ $1st$ question: If x is large, then $ y^{1/2}y'''$ and ...
1
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0answers
15 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
1
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0answers
8 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
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0answers
20 views

Changing form of differential equation

How would you change the differential equation: $3xy^{''}+y^{'}+12y=0$ into a form where the coefficient of the first term is 1? leaving just $y^{''}$ I should probably know how to do this, but I'm ...
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1answer
22 views

Limit of a continuously differentiable function that statisfies

Let $x(t)$ be a continuously differentiable for all $t>0$, and such that: $$\lim_{t\to \infty}[x'(t)+x(t)]=\alpha$$ I need to show that $\lim_{t\to \infty}x(t)=\alpha$ My goal is to show that ...
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0answers
15 views

Proving that the solution to the IVP exists given a condition on the right hand side

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$. Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$. In order ...
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0answers
10 views

Interval of solutions of this differential equation

For the following differential equation initial value problem, $y' = \frac {-t + (t^2 + 4y)^{1/2}} {2}$, $y(2) = -1$ the interval of t for which the solution $y_1 = 1 - t$ is valid is $t \ge 2$ and ...
0
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0answers
14 views

Limit of Solution of an ODE

Consider the following differential equation: $y^{'}(t)=g(y)$ where $g$ is a continuous function from $\mathbb{R^n}$ to $\mathbb{R^n}$. Assume that $y(t)$ is a solution to the previous ODE. Suppose ...
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1answer
16 views

Projectile Motion using cos and sin theta???

Golfball is struck to clear a tree 20m away and 6m high at an angle of elevation of 40degrees. Find the speed of the ball when it leaves the ground. I've created my displacement equation with i and j ...
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3answers
31 views

How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
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0answers
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Analyse existence and uniqueness for solutions of BVP [on hold]

$$u''-u+λ \arctan (ux^2) = 0,\: 0≤x≤1$$ $u(0) = 0,\: u(1) = 1, \:u ∈C^2([0,1])$ Here $λ$ is a real number and all functions are real-valued. What can be said for different values of $λ$?
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1answer
18 views

Why existence of Lyapunov function implies Lyapunov stability at the equilibrium point

Why existence of Lyapunov function (locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite) implies Lyapunov stability (i.e for any ...
1
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1answer
25 views

How to solve Laplace initial value problem

$$ y''+36y = f(t) $$ $$ f(t) = \begin{cases} 1, & \text{0 ≤ t < 8} \\ 0, & \text{8 ≤ t < ∞} \end{cases} $$ $$ y(0) = 0 $$ $$ y'(0) = 1 $$
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0answers
11 views

On a cauchy problem (differential equation system) .

I am approaching the theory of these kind of problems but I am missing an example. I am tasked to solve: $$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( ...
1
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1answer
15 views

Solution to Differential Equation $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$

I'm trying to solve the following Differential Equation: $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$ The unknown function is $w(x,y,z)$. The ...
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0answers
8 views

the solution to $1$st order linear nonhomogeneous equation (leading coefficient may vanish)

I'm working on the $1$st order linear nonhomogeneous equation $c_1(x)y'+c_2(x)y=f(x)$. I know the homogeneous solution is $h(x)=e^{-\int_ a^x c_2(t)/c_1(t) dt}$ and then $y(x)= e^{-\int_ a^x ...
0
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1answer
34 views

Specific form of differential equation

Suppose the function $$ f(x)=p(x)\,\mathrm{e}^{q(x)}, $$ is the solution to a differential equation. From which family of differential equations would $ f $ arise?
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1answer
21 views

Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
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0answers
11 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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0answers
10 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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0answers
21 views

Convergence to a fixed point

When the following system is given: $x(k+1)=r-rx(k)$ where $r>=0 $ is a parameter Can someone explain why the fixed points are given by: $x(k+1)=x(k)=x^*$, so $x^*=\frac{r}{1+r}$? and how to ...
0
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2answers
24 views

Second Order Linear Differential Equations

The question is that find the general solution of differential equation y"-2y'+y=e^x i know that y(c)=Axe^x + Be^x then let the f(x)=e^x , so y=pxe^x as f(x) is in the complementary function. so ...
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2answers
26 views

$\lim\limits_{t\to\infty}t-x(t)=0\ ?$

Let $\displaystyle\cases{ x'=\frac{t-x}{1+t^2+x^2} & \cr x(1)=1 }$ be the Initial value problem, prove or disprove $\lim\limits_{t\to\infty}t-x(t)=0$ We've already proved that: for $t>1, ...
2
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1answer
9 views

Help with Implicit Differentiation: Finding an equation for a tangent to a given point on a curve

When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book. The problem is ...
1
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2answers
25 views

Find the limit and differential equation

We have the following equality: $$ f(x + \Delta x) = f(x) + a \Delta x \, f(x) - 10 \, b \Delta x $$ with a & b constants. If we take $\lim_{\Delta t \to 0}$ , we get a differential equation. My ...
1
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1answer
10 views

Second-order Van der Pol to First Order System

I'm trying to figure out how my professor arrived at the following first order system for the Van der Pol equation $x''+ c(x^2-1)*x' + x = 0$. It's supposed to be equivalent to the first-order system ...
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0answers
31 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
1
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1answer
28 views

Solution to differential equation $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$

I'm trying to solve the following differential equation: $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$ here $g(x,y,z+h)$ is a known function that however i will leave ...
0
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1answer
15 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
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0answers
11 views

A basic doubt on Lyapunov function

Let $V$ be a Lyapunov function. Let $\epsilon > 0$ be any number such that the set $|y| < \epsilon$ is in the open set on which $V$ is defined. Then Hartman claims the following : For any ...
2
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1answer
24 views

Solving the ODE $(x^2 - 1) y''- 2xy' + 2y = (x^2 - 1)^2$

I want to solve this ODE: $$(x^2 - 1)y'' - 2xy' + 2y = (x^2 - 1)^2.$$ I found out that $y_1 = x$ and $y_2 = x^2+1$ are solutions of the associated homogeneous equation, $x$ by inspecting, and ...
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0answers
14 views

Periodic solutions and critical points

I was going through a lecture, and for an ODE: $x' = x(5-x-2y), y'=y(-6x+x+3y)$ Which has critical points at : $(0,0) (0,2) (3,1) (5,0)$ My professor posed the question as to why the periodic ...
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1answer
28 views

Chain rule and x =cos(θ) substitution on the Legendre ODE

I am having difficulties with a question where I am required to use the chain rule, and then use the substitution x = cos(θ), on the Legendre differential equation, which is $$(1-x^2)y'' - 2xy' + ...
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2answers
46 views

Solve differential equation, $x'=x^2-2t^{-2}$

Solve differential equation: $x'=x^2- \frac{2}{t^2}$ Maybe is it sth connected with homogeneous equation? I have no idea how to solve it.
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3answers
37 views

Solve $2xdy+(x^2y^4+1)y dx=0$ [on hold]

Solve differential equation: $2xdy+(x^2y^4+1)y dx=0$ Hint: use: $y=zg(x)$ Any ideas?
2
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3answers
70 views

Solve y''' = y with given conditions?

I'm given the differential equation: $$y''' = y$$ which solves to: $$y(x) = c_1e^x + e^{-x/2}\left(c_2\cos\left(\frac{\sqrt{3}x}{2}\right) + c_3\sin\left(\frac{\sqrt{3}x}{2}\right)\right)$$ But I'm ...
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0answers
15 views

How to find the solution of second order equation

I have a question that need your help. Given a $x(t)$ is a second-order Markov process: $$\frac {dx}{dt}=x_2+\omega$$ $$\frac {dx_2}{dt}=-2x-2x_2$$ with white noise $\omega$ is $(0,2)$. How to find ...
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1answer
27 views

STPM CHAPTER 6 DIFFERENTIAL EQUATION [on hold]

dy/d×= y(2×+y)/×(y-×) Given that y=2 when ×= 1
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differential forms and index

The other question i can´t solve is this, If $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$. ...
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0answers
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Difference Equations and their applications

What are some interesting applications to difference equations? I've learned about first and second order difference equations, first order systems, different kinds of equilibrium solutions (locally ...
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2answers
10 views

Using method of undetermined coefficients

Here's the equation I'm working with: y''+y'-2y = 3e^(x)+4x I want to use the method of undetermined coefficients for this equation. The logical choice for our guess would be y = Ae^(x)+Bx+C. ...
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0answers
31 views

1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
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0answers
18 views

The Adjoint Equation

I have a simple question about the adjoint equation for second order linear differential equation. Given an equation of the form $$P(t)y'' + Q(t)y' + R(t)y = 0$$ Let $u(t)$ be an integrating factor ...
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1answer
12 views

Stochastic Differential equation, expectation and variance

The process is given by $$dU_t=-\gamma U_t\mathrm{d}t+\sigma\mathrm{d}X_t$$ where $U_0 = u$ and $\gamma, \sigma$ are constants. Can you help me out to solve the equation for $U_t$ and find the ...
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0answers
15 views

A Bernoulli Differential Equations Problem [on hold]

Please, can anyone help me in solving this Bernoulli's Differential Equation. $(xy+x)dx=((x^2)(y^2)+(x^2)+(y^2)+1)dy$
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0answers
8 views

Question on stable manifolds

If $x\in M$ is a hyperbolic fixed point of a diffeomorphism $\phi:M\to M$, then the stable manifold $$ W^s=\{y\mid \lim_{n\to\infty}\phi^n(y)=x\} $$ is the image of an injective immersion $$ ...
0
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0answers
19 views

Differential equation $tx'= \sqrt{t^2-x^2}$ [on hold]

$tx'= \sqrt{t^2-x^2}, |x|\le|t|$ Is it sth like Clairaut equation?
0
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1answer
21 views

prove that $f(x,y) = x^2+y^2$ is continuous on rectangle R.

where $R = \{(x,y): |x|, |y| \leq \frac{1}{\sqrt 2} \}$ I am trying to use picard's theorem so I have to prove that f is continuous on R and that it's lipschitz continuous. How would I do this? I ...
1
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1answer
23 views

Determinant of solution of linear equation

Is there a direct way or method to know if the solution to a linear ODE is invertible? I mean, let $A(t)$ be a ($n$ times $n$) matrix and denote by $X(t)$ an unknown Matrix (of the same dimensions) ...
0
votes
2answers
37 views

Solve the initial value problem $u'(t)=u^2(t)+t,\;u(0)=1$

How can I solve the following initial value problem: $$\begin{cases}u'(t)=u^2(t)+t\\u(0)=1\end{cases}$$ This is a first-order nonlinear equation. The only method I know, to solve such an equation, is ...