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

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71 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as ...
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139 views

Linear first-order equation: $xy'+(1+x)y=e^{-x}\cdot\sin(2x)$

Question: following first-order equation $$xy'+(1+x)y=e^{-x}\cdot\sin(2x)$$ Thankful.
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1answer
230 views

Lipschitz continuity and Picard iteration

Let $F : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying a Lipschitz condition. Using the Picard iteration, I'm trying to show that the solution of any initial value problem for ...
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2answers
193 views

Differential Equation $y'-P(x)y=Q(y)$

Solve: $$(xy^4+y).dx=xdy$$ I tried but it ended into $$y'-P(x)y=Q(y)$$ Had it been $Q(x)$ , I would have been able to solve.[Lenier D.E.]. But how to solve this one?
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86 views

wronskians of two pairs of linearly independent solutions to second order homogeneous ODE

Let $p(x)$ and $q(x)$ be two continuous functions on an interval $(a, b)$ and suppose that $(y_1, y_2)$ and $(z_1, z_2)$ are two pairs of linearly independent solutions to the ODE $y'' + p(x)y' + ...
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57 views

Solution to first order differential equation

How do I argue that the following IVP has no meaningful solution? $$\frac{dx}{dt}=\sqrt{(x^2-t)},x(1)=0$$ The basic condition for existence of solution is that $\sqrt{(x^2-t)}$ is differentiable and ...
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84 views

Boundary-value problem in differential equations

Consider the problem: $$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$ Find the eigenvalues. How should one proceed about this problem? I am complete ...
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72 views

Simplify the nonlinear system of dynamic equations

I am working with a set of nonlinear dynamic equations that Mathematica has problems with solving. It is of the form $$ f_1(x_{t+1},y_{t+1},x_{t},y_t) = g_1(x_{t+1},y_{t+1},x_{t},y_t),\\ ...
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296 views

Differential equations notations confusion

Given these differential equations: $\frac{d^2x}{dt^2} = 2\Omega\frac{dy}{dt}\sin(\lambda) - \frac{g}{L}x$ $\frac{d^2y}{dt^2} = -2\Omega\frac{dx}{dt}\sin(\lambda) - \frac{g}{L}y$ Now making the ...
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2answers
107 views

Stuck at a differential equation, particular solution

The problem is: $y'' + 4y' + 3y = (4x-2).e^{-3x}$ with conditions $y(0)=2$ & $y'(0)=0$ I first find the characteristic polynomial $p(r) = (r+3)(r+1)$ which gives me the homogeneous solution $yh ...
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228 views

Legendre Polynomial Sum

The question has two parts: Does the sum of the Legendre Polynomials from $L=0 \to \infty$ evaluated at $x=0$ converge? What about the sum of the squared Legendre Polynomials at the same point?
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89 views

Trick in integration with Taylor expansion

I am struggling with the expression of the LHS of the following equation. The RHS is just the Taylor expansion of the first function around point y and the differentiation wrp to the argument y. ...
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1answer
37 views

Change of variables to find the interval

By the change of variables, if I let $\frac{z^b}{e^z}\frac{d}{dz}$=$\frac{d}{dw}$, i.e., $\frac{dw}{dz}$=$\frac{e^z}{z^b}$ for $0<z<\beta$, then how to compute w(z)? Does $0<z<\beta$ ...
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48 views

Homogeneous equation2

Please help me to solve this question. Thanks Question: resolvent Homogeneous equation $=> (x\sin(\frac{y}{x}))dy+(x-y\sin(\frac{y}{x}))dx=0$ My Attempt: $$ v=\frac{y}{x}\\ y ′=\frac{(y ′ ...
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235 views

Gronwall Lemma.

Consider $x'(t)=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2(x_2),-2(x_1)-x_2)$. I need to show that for two solutions $x(t)$ and $y(t)$ of the above differential equation we have: $\lVert ...
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409 views

What's The Degree Of Given ODE?

What is the degree (order looks 2) of the following ODE (and whats the way to find it?) ? $$k{(y'')}^2 = (1+ {(y'')}^2)^3 $$
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90 views

Flow of D.E what is the idea behind conjugacy?

I got some kinda flow issue, ya know? well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the ...
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59 views

separable equation1

Does this solve the Impartible Equation is correct? Impartible Equation: $(x+1)y'y=y^2-1$ solve: $y'=y^2-1/y(x+1)$ $f(x,y)=y^2-1/y(x+1)$ $y^2-1/y/(x+1)/1$ $f(x,y)=(y^2-1)/y/(x+1)/1$ ...
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229 views

Variation of parameters

I solved the differential equation $y'+ky=e^{rt}$ and found $y=\frac{e^{rt} }{r+k}+\lambda e^{-kt}$, for $r+k\neq0$ But I need to solve the missing case $r=-k$. I was thinking of using the method of ...
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92 views

How to calculate $\int \dfrac {e^{t}} {1+t}dt$

The original is slove $\dfrac {df} {dt}+\dfrac {t} {1+t}f=at(ie.a*t)$, IC:$f\left( 0\right) =f_{0}$ I used method of variation of constant and get a indefinite integral $\int \dfrac {et} {1+t}dt$, ...
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183 views

Method of Undetermined Coefficients

I am trying to solve a problem using method of undetermined coefficients to derive a second order scheme for ux using three points, ...
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39 views

Necessary conditions for transforming a system of O.D.Es to a single O.D.E.

Considering the system: $$u' = a(x)u + b(x)v$$ $$v' = c(x)u + d(x)v$$ I transform it to the second order O.D.E.(please check me on this because I might have mistakes): $$\displaystyle u'' - \left ...
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172 views

Solve the integral equation

$$y(x) = 2 + \int_8^x (t-ty(t))dt$$ I am having a very hard time doing this problem. (i) Solve the separable differential equation $$y'(x) = x − xy(x)$$ to get $$y(x) = 1 + c \cdot e^{−x^2/2}$$ (ii) ...
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346 views

Find the solution to the given differential equations using variation of parameters

So the equations is $y''-5y'+6y=g(t)$ I found the characteristic equation to be $r^{2}-5r+6=0$ which factors to $(r-3)(r-2)$ So the fundamental set of solutions to the homogeneous equation is: ...
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625 views

finding an integrating factor for the nonlinear non-autonomous ODE $ (xy)y'+y\ln y - 2xy = 0 $

I though i might refresh my ODE knowledge a bit and decided to try the following exercise from "Advanced mathematical methods for scientists and engineers" by C. Bender and S. Orszag Solve the ...
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2k views

Solving System of Differential Equations with initial conditions maple

I've been asked to solve a system of differential equations using maple (for practice, as it is solvable by hand...), but I seem to be running into goop with syntax... d/dt r(t) = -lambda_r*r(t) + ...
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68 views

Solve the Euler's equation $x^2y^{\prime \prime}+3xy^{\prime}+10y=0$

Solve the Euler's equation $$x^2y^{\prime \prime}+3xy^{\prime}+10y=0$$ For $x>0$, I use the transformation $z=ln x$ to get a homogenous equation with constant coefficient and then I manage to solve ...
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63 views

Why are the periodic solutions of conservative unidimensional systems symmetrical regarding the $x$ axis?

Consider the equation $$ x''=F(x) $$ which is equivalent to $$ \begin{array}{l} x'=v\\ v'=F(x) \end{array} $$ I have already shown that all the equilibrium points of the system are on the $x$ axis, ...
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866 views

Determining whether an IVP has a unique solution if uniqueness theorem doesn't hold

I have the following initial value problems: $$ \begin{cases} y' = y^{\frac{1}{2}}\cos(x) \\[6pt] y(-2)= 1 \end{cases} $$ $$ \begin{cases} y' = y^{\frac{1}{2}}\cos(x) \\[6pt] y(5)=-8 \end{cases} $$ ...
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149 views

Stuck on homogeneous linear equation $y' ={ {x^2+xy+y^2} \over x^2}$

Given this first-order linear equation: $y' ={ {x^2+xy+y^2} \over x^2}$! I have to show first that it is homogeneous. I divide numerator and denominator of RHS by $x^2$ to get $dy/dx=1+{y \over ...
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115 views

Verifying a (convolution) solution for ODE

I'm having trouble in verifying that $f\star \phi$, $\phi = \frac{1}{2} e^{-|x|}$ is a solution for $u -u''=f$. Please help me.
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3answers
192 views

Differential equation initial value problem - hard!!

I have been asked to solve $x' = t/(1 + t^2) - x(t/(1+t^2))$ and determine the maximal interval where the solution exists. I have tried to solve this in many different ways but must be using the ...
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477 views

What is the particular solution of $y'' - 4y' + 3y = 2t + e^t$

$y'' - 4y' + 3y = 2t + e^t$ Usually this will = 0. So I would just need to find the characteristic equation and factor it. In this case its $ \ne 0 $ so what do I do? *What is the difference between ...
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78 views

Uniqueness Theorem query.

Is there any proof which would show that the uniqueness theorem does not imply complete uniqueness of a ODE solution? Would it have any relation between the interval where uniqueness is assumed to ...
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2answers
55 views

Solution to the following differential equation

I am trying to find the solution for the differential equation $\frac{dz}{dt}$ = $z^{\alpha}$ for some $0<\alpha<1$. Can anyone help me out here!! Thanks in advance
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2answers
306 views

Solve a system of time-independent ODE's with vector constants

I have to solve numerically this set of Ordinary Differential Equations $$ \frac{dx_1}{ds} = \frac{1}{x_1} \left[x_2 \left(a + \frac{x_2}{s}\right)-\alpha x_1 z\right]$$ $$ \frac{dx_2}{ds} = ...
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44 views

$L$ elliptic diff op $\implies$ singsupp$(u)\subseteq$singsupp$(Lu)$ for distributions $u$?

If $L:D'(\mathbb{R}^n)\to D'(\mathbb{R}^n),n\in\mathbb{N}$ is a weakly elliptic, linear differential operator with constant coefficients then for every $\Omega\subseteq\mathbb{R}^n$, and for all $u\in ...
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40 views

Set of differential equations

I've just begun to study ODE and I have to solve this set of equations: $$\frac {d v_x}{dt}=\omega v_y$$ $$\frac {d v_y}{dt}=-\omega v_x$$ I have made these steps: $$v_y=\frac{1}{\omega} \frac{d ...
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26 views

Why doesn't this DE's interval of definition include negative $x$ values?

Why is the interval of definition for $xy' + 4y = x^3 - x$ given by $0 < x < \infty$ instead of $-\infty < x < 0 \; \cup \; 0 < x < \infty$? When I solved it I got $y = ...
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87 views

Large system of ordinary differential equations

I am trying to find a large system (>20) of coupled ordinary differential equations in order to approximate them numerically on the computer and check the efficiency and effectiveness of various ...
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2answers
66 views

Differential equation variation of parametres

Find the general solution of $y'' + \dfrac{7}{x} y' + \dfrac{8}{x^2} y = 1, x > 0$ I don't even know how to solve the homogeneous version because it involves variables... Does anyone know how ...
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1k views

Differential equation on half-life isotope decay

I'm currently working on some Differential Calculus, and I'm having a bit of trouble with the following question. The half-life of an isotope is 150 years. Use this information to determine ...
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617 views

Continuity of solutions of ODEs with Dirac delta forcing

The problem is : Find $y$, if $y''-4y=\delta(x-a)$, where $\delta$ is the Dirac delta function, and $y$ is bounded as $|x|\rightarrow\infty$. The solution goes like this: For $x<a$, ...
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197 views

Differential to Difference equation with two variables?

For the following information : $$\frac{dx}{dt} = -10x+3y$$ $$\frac{dy}{dt} = 2$$ How do I convert this to a difference equation ?? I want to use a simple discretisation technique (first order) ...
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143 views

Analytic Function without Power Series

If $$f(x) =\sum_{n=0}^{\infty}x^ n$$ Then Determine the function $f(x)$. Discuss the domain of $f(x)$. Discuss the domain of the derivative of $f(x)$. Thanks!
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2k views

Solving a second-order nonlinear ordinary differential equation

Let's start with equation with two parameters $$y=a x^b.$$ Then we calculate $y'=a b x^{b-1}$ and solve $a=y/x^b$ from original. Substitute that to the derivative and $$y'=b \frac{y}{x}.$$ Then ...
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23 views

Satisfying a Differential Equation

So, $y=2cos(kt)$ Therefore, $y'=-2sin(kt)k$ Thus, $y''=-2cos(kt)k^2$ Plugging this into $4y'' = -16y$ ... $4(-2cos(kt)k^2)=-16(2cos(kt))$ I've simplified it this far: $(cos(k)k^2+4cos(k))=0$ ...
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70 views

Definition of purely oscillatory

This is question about a term whose definition I can find anywhere. I am given to solve a differential equation and one of the questions asks to show that the solution (we are given initial data) is ...
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292 views

differential equation with distributions

I'm stuggeling with this differential equation: $T'+T=0$ Where $T$ is distribution. I found solutions in form: $\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to ...