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

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Lyapunov Functions and Basins of Attraction

\begin{align} x' &= -x^3 + x^5 + (x^4)(y^5)\\[.7em] y' &= -8y^3 + y^5 - 10(y^4)(x^5) \end{align} $(0,0)$ is obviously a critical point of the system, and we are given that it is ...
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43 views

Convolution with Heaviside function (integration)

To clarify notation, I use $u_n = 1$ when $x>n$, and $0$ otherwise. I am having troubles with the following convolution/integration: $u_2(t) \ast sin(\sqrt{2}t) = \int^t_0u_2(\tau) \cdot ...
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27 views

Differential equation with boundary conditions

Consider, for all $-1 < x < 1$, the eigenvalue problem $$u''+\lambda u = 0$$ with boundary conditions $$u(1)+u(-1)=0 \quad \text{and} \quad u'(1)+u'(-1)=0.$$ Prove that for $u,v \in ...
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33 views

Nonhomogenous Second Order Ordinary Differential Equation issues.

I had some trouble on an exam recently with the particular solution of a Nonhomogeneous Second Order Ordinary Differential Equation. So, the problem was: \begin{cases} y''+4y=\sec(2t) \\ ...
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5answers
84 views

What does the solution of a PDE represent?

So I took a course in PDE's this semester and now the semester is over and I'm still having issue with what exactly we solved for. I mean it in this sense, in your usual first or second calculus ...
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65 views

Why do we need two linearly independent solutions for 2nd order linear ODE

Let we have a second-order homogeneous linear ODE with two initial conditions. $y''+ p(x)y'+q(x)y=0$ $y(x_0)=K_0$ and $y'(x_0)=K_1$ Why do we need two linearly independent solutions to satisfy the ...
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40 views

Is there a general coordinate transformation perserving the components of an Euclidean metric?

In the Euclidean space (or Lorentz spacetime, if you are interested in relativity), there is one orthonormal coordinate system $\{x^\mu\}$ such that the distance squared is given by ...
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22 views

Sturm-Liouville and Continuity

If you have a differential equation $$ y'' + R(x)y' + (Q(x) + \lambda P(x))y = 0$$ on some interval $(a,b)$ why must the coefficient functions be continuous on this interval? My guess is that if you ...
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1answer
21 views

diffrential equation - prove that any solution is not intersecting with the x axis

I'm trying to tackle the following question: Let $\displaystyle y'+p(x)y=0$, such that $\displaystyle p(x)$ is continuously differentiable and bounded. Show that every solution which is not ...
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49 views

Two differential equations - separation of variables, Gronwall's lemma

1) How to solve $xyy' = \ln x$ without separation of variables? That's what I am asked to do. $$yy' = \frac{\ln x}{x} \Rightarrow \frac{y^2}{2} =\frac{\ln^2 x}{2} + C \Rightarrow \dots$$ ...
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38 views

Problem with solution for non homogeneous differential equation

I am trying to solve this non-homogeneous equation, but my answer is a bit off for some reason: $$y''-4y=x^2+3e^{2x}$$ I had $$y_c=c_1e^{2x}+c_2e^{-2x}$$ Then for $y_p$, I used ...
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36 views

inhomogenous Differential equation

I have this inhomogenous equation. (*) $y'' + 5y' - 6y = 14e^x$ I solve the Homogenous part and get $\lambda_1 = 1 $ $ \lambda_2 = -6 $ $Ce^x + De^{-6x}$ Particular solution: I try here to ...
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35 views

Question in Differential Equation (How to proceed?)

By using the substitution $y=vs$, show that the general solution of the first order homogeneous differential equation $(x+y)\frac{dy}{dx}=y-x$ in the case where $x>0$ is given $\tan ...
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53 views

Plotting the graph of systems of ODE

The eigenvalues and eigenvectors of a matrix A are given. Consider the corresponding system $x' = Ax$. (a) Sketch a phase portrait of the system. (b) Sketch the trajectory passing through the ...
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20 views

Self-Adjoint Linear Differential Equation

Given a function $a(x)$, consider the differential equations $y'=a(x)y$ and $y'=-a(x)y$. Let $\phi(x), \psi(x)$, respectively, be the unique solutions of these which are equal to $1$ at $x=0$. I want ...
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96 views

Solving Systems of Linear Differential Equations by Elimination

For a homework problem, we are provided: $\frac{dx}{dt}=-y + t$ $\frac{dy}{dt}=x-t$ Putting these into differential operator notation and separating the dependent variables from the independent: ...
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3answers
43 views

What initial value do I have to take at the beginning?

In my lecture notes there is the following example on which we have applied the method of characteristics: $$u_t+2xu_x=x+u, x \in \mathbb{R}, t>0 \\ u(x,0)=1+x^2, x \in \mathbb{R}$$ $$$$ ...
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68 views

Find extremal of the functional

$$J(u)=u(0)+0.5 u(0)^2 + \int[u'(t)]^2 dt$$ $u(0)$ unsepecified $u(1)=2$ I know it solved by e-lagrang equation But i dont know which one How i can concered with $u(0)$ which is out of integral ...
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58 views

How can we get a contradiction?

How could we show that the problem $$u_{tt}(x, t)-u_{xt}(x, t)=0, x \in \mathbb{R}, t \in \mathbb{R}, \\ u(x, -x)=0, x \in \mathbb{R}, \\ u_t(x, -x)=x, x \in \mathbb{R}$$ doesn't have any smooth ...
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3answers
57 views

Series solution to first order differential equation

I need to find a series solution to the following simple differential equation $$x^2y'=y$$ Assuming the solution to be of the form $y=\sum a_nx^n$ and equating the coefficients on both the sides, all ...
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60 views

How to find the general solution to this ODE…

Q: Find the general solution to the ODE:$$t^2y''-ty'+y=0~~~~~~~~(*)$$given that $y_1=t~~$is a solution. My intuitive solution: Let $x=ln(t)$, then $y'=\frac{dy}{dx}\frac{1}{t}$ and ...
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2answers
20 views

DE undetermined coefficients

De problem $$x'' + 196x = 40\sin (14t)$$ I have the general solution, I'm just struggling with the algebra with the undetermined coefficient. I set my guess as: $At\cos (14t) + Bt\sin (14t)$, took ...
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43 views

Historical Development of PDEs

I would be really thankful if someone could tell me good references giving development of techniques of Solving PDEs why such equations are important. Regards, Harish
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39 views

solution to a 2nd order differential equation

I trying to solve the following differential equation. $$(x^2-3)y''+6xy'+4y=0$$. Should I use the power series solution or the Method of Frobenius?
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22 views

laplace transformation of a function using definition

I want to find the laplace transformation of $x^ne^{ax}$ using the definition. I'm stuck with the integral. How shall I proceed the integral and find the final answer with $n$?
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26 views

Method of characteristics- $u=$ constant or $u = f(y)$?

Say $u(x,y)$ is a function of $x$ and $y$ and suppose we have the following pde - $u_x - u_y = 0$ This equation has the following characteristics - $\frac{dx}{1} = \frac{dy}{-1}$ $\frac{du}{dx} = ...
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63 views

solution to a specific differential equation

Does this equation have a known solution given a starting value $\phi_1$? $$\phi_{n}\prod_{j=1}^{n-1}(1-\phi_{j}^2) + \phi_{n-1}\phi_1=0$$ Background: I am trying to derive the partial ...
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31 views

Fundamental Set of Solutions and Wronksian

Explain why the pair of functions $y_1(x) = x$ and $y_2(x) = \sin(x)$ cannot form a set of fundamental solutions to a second order homogeneous differential equiationon the interval $(-1,1)$. ...
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69 views

analitycs solutions to the equation $f'(x)=f(x)f(x-1)$

As the title says I'm serching for functions ($C^n$ or analitycs $f$) that satisfies $f'(x)=f(x)f(x-1)$ some details: I've come at this equation after looking for a function $g$ satisfying for ...
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45 views

Have I obtained the proper solution to this PDE?

I'm a little stuck on this. Consider $ u_t -(1+t^2)u_x = \phi(x,t) \quad u(x,0)=u_0(x)$ Via the method of characteristics, the total derivative of $u(x,t)$ is $$\frac{du}{dt} = \dfrac{\partial ...
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59 views

Ordinary differential equation of order 2, degree 3.

I wonder, how could I solve an equation like this? $y''+(y-\frac{y^3}{6})=0$ Or more general: $y''+\frac{g}{l}(y-\frac{y^3}{6})=0$ Any hint?
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56 views

How to bound error when approximating ODE

I have a question regarding how to bound the error, if one changes the "right hand side" of an ODE. For example, the equation of a simple pendulum in polar coordinates is something like ...
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36 views

Differential equations - bounded dynamical system

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $C^1$-function and let $I_{x_0}=(a,b)$. Assume that there exist $M>0$ such that $|\varphi(\cdot,x_0)|_{[0,b)}|\le M$, where $\varphi(t,x)$ is dynamical ...
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31 views

Why is the ODE linear?

I am looking at initial value problems for ordinary differential equations. Let $a,b, \ a<b, \ f: [a,b] \times \mathbb{R} \to \mathbb{R}$ function and $y_0 \in \mathbb{R}$. We are looking for a ...
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39 views

Finding the analytical solution to this second order ODE

I need to find the solution to; $$y''= \frac 2xy' - \frac {2}{x^2}y - \frac 1{x^2},\ y(1)=0,\ y'(1)=1 $$ By observing that the first two terms on the RHS exactly for the derivative of some function; ...
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35 views

A differential equation with a complex variable

Consider the differential equation $$u^{\prime}=p(z)u\quad\text{for}\ u\in{}D,$$ where $p$ is a complex-valued continuous function on the open disc $D\subset\mathbb{C}$. If $z_{0}\in\overline{D}$, ...
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39 views

Integration of a differential that is a function of itself.

Is there an explicit solution, or even a good way to numerically integrate, a relation such as: $$\frac{dT}{dt}=-C_1T^{C_2}$$ $C_1$ and $C_2$ are constants. $C_2$ is NOT $1$.
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66 views

Prove General Solution To System Of ODE.

Prove that $\alpha e^{\lambda t} \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} + \beta e^{\lambda t} \begin{pmatrix} t \\ 1 \\ \end{pmatrix}$ is the general solution of $X' = \begin{pmatrix} \lambda ...
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32 views

Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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50 views

Find all solutions of the 2'nd order ODE: $x'' + x =0$.

Find all solutions of the 2'nd order ODE: $x'' + x=0$. I've found by inspection of $x (t)=e^{at} $ that $\cos t $, $\sin t $ are solutions. How do I see, that all linear combinations of these are all ...
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42 views

On which interval in x is the solution defined?

Here is the differential equation. $$ y' = \frac {2x}{1+2y} $$ Given $$ y(2) = 0 $$ So as I am solving this I get to $$ y+y^2 = x^2-4 $$ I am not sure what the intuition is behind using the ...
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57 views

Legendre polynomial to show identity, can't spot mistake

Using Legendre polynomial generating function \begin{equation} \sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}} \end{equation} Or $$ P_n(x)=\frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] ...
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19 views

Linearization of autonomous DE.

I have been working through this PDF about DEs. There was the logistic equation $$\dot{y}=k_0y(1-y/p),$$ where $k_0$ and $p$ are constants. Then a new variable is introduced $y=u+p$. Then ...
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46 views

Basic question about terminology, notation and definitions in calculus

When reading stuff about differential equations I'm coming across some strange (for me) notations/terminology. For example, when coming across something like this: $$\frac{dy}{dt}=f(y,t)$$ or ...
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30 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} + ...
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48 views

Gronwall's Lemma type problem

I have a function $X(t)\geq 0$, with initial condition $X(0)=X_0\geq 0$ and constants $\alpha < 0$, $\beta > 0$ and $\gamma <0$ such that $$\frac{d}{dt} X(t)^2 \leq \alpha X(t)^2 + \beta ...
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98 views

Mass spring system with friction

In class, we derived the displacement for a mass on a spring without friction as $$x=x(t) = x_o\cos(\omega_ot) + \frac{v_o}{\omega_o}\sin(\omega_ot)$$ We derived this equation from conservation of ...
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1answer
23 views

What can I imply from $\epsilon^\prime(x) \le -x\epsilon(x)$?

Imagine, that I can prove $$\epsilon^\prime(x) \le -x\epsilon(x)$$ for a function $\epsilon:\mathbb R \rightarrow \mathbb R_0^{+}$. Does this imply $$\epsilon(x) \le c e^{-\frac{x^2}{2}}$$ whereby $c ...
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39 views

Solving the functional $\min \int_0^1y^2y'^2\;dx,\;y(0)=0,\;y(1)=1$

I'm trying to solve the following problem: Determine smooth extremums in $$\min \int_0^1y^2y'^2\;dx,\;y(0)=0,\;y(1)=1$$ by (a) using the fact that the functional does not contain ...
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60 views

Fourier Series and differential equation with epsilon

Happy New Year! I am stuck for days on expressing the solution of a differential equation using Fourier series. The question is: Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha ...