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

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1answer
55 views

Solving Non-homogenous System (repeated case)

I have the following system. $\vec{x^{'}}(t)=\begin{pmatrix}4&-2\\8&-4 \end{pmatrix} \vec{x}+ \begin{pmatrix}t^{-3}\\-t^{-2}\end{pmatrix}$ I get $\lambda=0$ and the eigenvector of $$4x_1=2x_2 ...
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2answers
89 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
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1answer
25 views

Estimation higher order

Consider non-dimensional differential equation for the height at the highest point is given by \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} \log_e(1+\mu) \end{equation} $0<\mu\ll 1.$ ...
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2answers
76 views

Does a solution to the differential equation $y'=y$ exist? [duplicate]

What is the solution to this differential equation : $$f'(x)=f(x)$$ I'm very interested in this because if it have a solution this means that the slope of that function at a point $a_0$ is the height ...
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1answer
37 views

How to find I(t)?

I'm working with a SIS model for diseases. Where S stands for susceptibles, and I stands for infected. I have a situation that is modeled by the system: $$S'(t)=\frac{dS}{dt}=-\beta SI-\lambda S$$ ...
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2answers
239 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
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1answer
268 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
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1answer
76 views

Termwise Integration of Fourier Series

This is a question from Edwards and Penney 4th edition Differential Equations and Boundary value problems from section 9.3. Suppose that $f(t)$ is a piecewise continuous period $2L$ funtion. ...
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2answers
53 views

First order differential equation confusion

I have a differential equation $$y' + e^{y'}-x=0$$ that I have simplified like so $$e^{y'}=x-y'$$ $$\ln e^{y'}=\ln (x-y')$$ $$y'= \ln (x-y')$$ but I do not know how to solve this further to obtain the ...
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2answers
68 views

How many methods are there for solving repeated roots of differential equations?

I am studying differential equations and the book states there are many methods of finding the solutions of a differential equation that has repeated roots, but it only gives one method. D'Alembert ...
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2answers
73 views

Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$.

This is an exercise of my course of PDE: Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$. My Attempt A function $g$ is called harmonic if $\Delta ...
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1answer
30 views

Inverse matrix of solution (of system ode's)

How can I prove that if we have the system of ode's of order $n$ $$\frac{d}{dt}X(t)=A(t)X(t), \text{with}\quad X(0)=I_n$$ where $X$ and $A$ are $n\times n$ matrices, then ...
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3answers
333 views

How can $x^2y''-3xy'+3y=0$ be solved?

a) How does $x^2y''-3xy'+3y=0$ can be solved? I know how to solve for constant coefficients, but in this case they are functions... b) In which maximum interval there is a solution that confirms ...
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1answer
53 views

Solving IVP with delta function

Solve the IVP. $y^{''}+2y^{'}+2y=\delta(t-\pi); \;\;y(0)=1\;\;y^{'}(0)=0$ Here is what I did: I took the Laplace transform of the IVP and obtained the following $$(s^2+2s+2)Y[S]-s-1=e^{-\pi s}$$ ...
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1answer
68 views

Why do we use $x^{m_1}$ instead of $e^{m_1x}$ for general solutions to Cauchy-Euler equations?

When I learned to calculate the general solution to homogeneous linear differential equations, I was told that for say, $y''-y=0$, the auxiliary equation is $m^2-1=0\implies m^2=1\implies m=\pm 1$ ...
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1answer
67 views

Help finding the particular solution of a 2nd order ODE

I'm reviewing for an exam, and I'm puzzled as to how to find the particular solution of the following 2nd order ODE: y" - (2/t)y' + (2/t^2)y = 2 ICs: y(1) = 0, y'(1) = 0 I know that the ...
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1answer
47 views

Improper integral and the ordinary differential equation 2

How to prove that $$f(t)=\int_0^{\infty}\frac{e^{-tx}}{1+x^2}dx$$ satisfies $$f''(t)+f(t)=\frac1t ?$$
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1answer
89 views

Improper integral and the ordinary differential equation

How does one calculate a derivative from improper integral like this? $$\int_t^{\infty}\frac{\sin(x-t)}{x}dx\qquad\ $$ It's been said that this particular integral (as a function) satisfies the ...
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3answers
59 views

What is $y''$, where $y' = 1 - t + 4y$?

I'm having a brainfart while trying to solve a problem for differential equations that requires me to recall some Calculus. If I have $y' = f(t, y) = 1 - t + 4y$, what is $y''$? Do I just ...
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1answer
89 views

non-homogeneous constant co-efficient 2nd order linear differential equation

I am doing a perturbation theory question and am having trouble with the (seemingly simple!) differential equation method of undetermined coefficients... I have reduced my given system so that now I ...
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2answers
46 views

Given a solution to find a matrix

For $e^{At} = 1/2\begin{bmatrix}e^{2t}+e^{-t} & e^{2t} - e^{-t} \\ e^{2t}-e^{-t} & e^{2t}+e^{-t}\end{bmatrix}$ for all t $\in$ $\mathbb{R}$. how to find A?
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1answer
46 views

A inital value problem $x' = f(x), x(0) = 1$ that has more than one solution?

A inital value problem $x' = f(x), x(0) = 1$ that has more than one solution? is this possible? if so, could you show me an example?
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1answer
71 views

Help Determine the equilibrium points and bifurcation value(s) for this family of DE.

Consider $x' = -x^4 + 5ax^2 - 4a^2$ a) Determine the equilibrium points and bifurcation value(s) for this family of DE. First I let $y = x^2$ Then set $-y^2 + 5ay - 4a^2 = 0 $ >>> $\frac{-5a +- ...
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2answers
82 views

Eigenvalues and eigenvectors of an integral operator

We have the following integral operator $$ Ku(t)=\int_0^1 G(t,s)\, u(s)\, ds,\,\, u\in L^2[0,1], $$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq ...
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1answer
49 views

solve the system $x' = x^2; y' = y^2$

How do I solve the system $x' = x^2; y' = y^2$ I know this is completely decoupled. However, I forgot how to solve a system with nonlinear terms. Could anyone get me started on this?
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2answers
52 views

D.E: ${\large \frac{dx}{dt}}=(x-1)(1-2x)$

I don't know how can I prove that $t=\ln\left(\dfrac{2x-1}{x-1}\right)$ is the solution of the following Differential Equation: $$\dfrac{{\rm d}x}{{\rm d}t}=(x-1)(1-2x)$$
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1answer
89 views

Jordan canonical forms and deficiency indices

I'm solving a homework question that asks me to do the following: "List the five upper Jordan canonical forms for a $4\times 4$ matrix $A$ with a real eigenvalue $\lambda$ of multiplicity $4$ and ...
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1answer
80 views

Solve the equation $a(x)y' + b(x)y=c(x)$. What if $a(x)=0$ somewhere? Does it always blow up?

Solve the equation $$a(x)y' + b(x)y=c(x)$$ (Hint: integrating factor.) I believe that if we assume $a(x)$ is differentiable and nonzero on an interval $(\alpha, \beta)$, we can use the ...
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2answers
86 views

Numerical solution of $y′′+2y=−x$?

How to solve $y′′+2y=−x$ differential equation numerically. $y′(1)=0$ and $y(0)=0$ ?
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1answer
1k views

Change of variables of differential equations, and in particular, initial value problems

Consider the following task: Transform the given initial value problem into an equivalent problem with the initial point at the origin: $\frac{dy}{dt} = 4 - y^3$, $y(-1) = 2$ I have a feeling ...
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2answers
40 views

Prove that the average of $D_wD_wf(x_0,y_0)$ over all unit vectors $w$ is equal to $\frac{1}{2} \Delta f(x_0,y_0)$ for any smooth function $f$.

Here is a challenge problem from my math professor: Let $w$ be a unit vector in $\mathbb{R}^2$, and let $D_w$ denote the directional derivative with respect to $w$. Prove that for any smooth ...
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4answers
110 views

Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
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1answer
67 views

Deriving a differential equation

I have the following information: $$\mathrm{i})\;\frac{\partial\theta}{\partial t}=D\frac{\partial^2\theta}{\partial x^2}\qquad \mathrm{ii})\;Q=\int_{\mathbb{R}}\theta ...
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1answer
48 views

Taylor series expansion of $f(x)=\frac{sinx}{x-\pi}$ at $x=\pi$

I was solving it and on one step I need to find the 2nd Derivative of $f(x)$, I am getting -1/3, but according to book it's -1/6.Please help me out here.
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1answer
959 views

Kirchhoff's Voltage Law

I am given: Kirchhoff’s voltage law states that the sum of the voltage drops across an inductor, L dI/dt, and across a resistor, IR, must be the same as the voltage source, E(t), applied to the ...
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2answers
70 views

Finding general solution for a nonhomogeneous system of equations

I have a system of differential equations: $\begin{cases} x_1'=x_2+2e^t \\ x_2'=x_1+t^2 \end{cases}$ And I want to find the general solution for it. I started by finding the general solution for the ...
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2answers
36 views

I need a reference for: Existence and Uniqueness of a general ODEs with a linear operator

I'm looking for a reference of a theorem that establishes the existence and uniqueness of the following general ODE: Let $Q_n$ is a finite dimentional Hilbert space and let the operator $A:Q_n\to ...
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1answer
78 views

Solution of $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-6y=0$ using $D$ operator

I've been asked to solve the following differential equation: $$\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-6y=0$$ I know how to solve it taking the trail solution $y=e^{mx}.$ But is the following approach ...
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2answers
432 views

Stable/unstable equilibrium points

Consider: $$ \dfrac{dN}{dt} = -rN \left(1-\dfrac{N}{K_1}\right) \left(1- \dfrac{N}{K_2}\right), $$ where $r,K_1,K_2$ are constants s.t. $r>0 $ and $0 < K_1<K_2$. Find the the ...
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1answer
605 views

Number of arbitrary constants in the general solution of an ODE

Why is the number of essential arbitrary constants in the general solution of an ODE (ordinary differential equation) equal to the order of the ODE?
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105 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
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1answer
61 views

What does this equation mean?

I have an exam tomorrow and I was trying to solve my Homework questions. I am stuck at this question: Find the general solution of the equation $$tdy + ydt = 3t^3y^2dt$$ It was exactly written as ...
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2answers
231 views

Try to solve the following differential equation: $y''-4y=2\tan2x$

I am trying to solve this equation: $y''-4y=2\tan2x$ the Homogeneous part is: $$y_h=c_1e^{2x}+c_2e^{-2x}$$ and I get according the formula: $$C_1'e^{2x}+C_2'e^{-2x}=0$$ ...
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1answer
81 views

Using Laplace transform to solve an IVP

The original problem is: $y''+9y = e^{-t} \sin (t)$ and the initial conditions are: $y(0)=-1$ and $y'(0)=1$. I've taken the laplace of both sides and got: $Y(s^2+9) = \frac{1}{(s+1)^2 +1} -s +1$ ...
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1answer
44 views

Problem with solving non-linear differential equation.

Need some hints where to start with this non-linear differential equations. $$\ddot{r} = \dot{r} (\dot\varphi)^2 - \frac{2rk}{m}$$ $$\ddot\varphi=-\frac{\dot{r}\dot\varphi}{r}$$ Thanks in advance ! ...
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1answer
143 views

Change of Variables in differential equation

Given the equation $zZ''(z) + Z'(z) + \alpha^2Z(z) = 0$ use the change of variables $x = \sqrt{\frac{z}{a}}$ where $a$ is a constant to map the problem to the differential equation $Z''(x) + ...
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1answer
364 views

ODE reducible to homogeneous equation

Solve the following ODE $x'=\dfrac{2x-t+4}{x+t-1}$. My attempt at a solution: If I make the substitutions $x=y-k, t=s-h$ with $k,h$ constants, then $x'=y'$ and the original equation reduces to: ...
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3answers
240 views

Find a power series solution centered at 0 (Differential equations

Here's the problem: $$(x-1)y''+y'=0$$ This is the work that I've already done: $$y=\sum_{n=0}^{\infty}a_{n}x^n$$ $$y'=\sum_{n=0}^{\infty}(a_{n+1})(n+1)x^n$$ ...
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3answers
85 views

How to solve this initial value problem on $(-\infty, \ +\infty)$?

I've managed to solve the following initial-value problem on the interval $(0, +\infty)$: $$x y^\prime - 2y = 4x^3 y^{1/2} $$ with $y = 0$ when $x = 1$. The unique solution is $y = (x^3 - x)^2$. How ...
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1answer
114 views

Markov Chain Solution Eigenvalue

I am having trouble understanding how to solve for the state vector at time $t$ for a markov chain using matrix algebra. I have the following Markov Transition Intensity Matrix, for the states A, N, ...