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

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14 views

Another integral equation question

Suppose that $z = \int_{- \infty}^z f (y) d y$. If $f$ were continuous, we can differentiate both sides to get $f(y)=1$. But what if $f$ does not have to be continuous, is this still true or are there ...
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2answers
22 views

I can't find the general solution (Second Order Linear Equations)

I'm trying to find the general solution to: y'' + 4y = t^2 + 7e^t The actual problem wants me to find the initial value problem with y(0) = 0 and y'(0) = 2 but I'm confident that I can find the IVP ...
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1answer
14 views

Conditions for Deriving $R_0$ for SIR Model Using Survival Function Method

I'm taking a look at the SIR model given by the system of differential equations \begin{align} \frac{dS}{dt} & = - \beta S I \\ \frac{dI}{dt} & = \beta S I - \gamma I \\ \frac{dR}{dt}& ...
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2answers
32 views

Why don't we write general linear second order ODEs as $P(x)u''(x) + Q(x)u'(x) + R(x)u(x) = g(x)$ instead of $u''(x) + q(x)u'(x) + r(x)u(x) = g(x)$?

The general form of the second order linear ODE is: $$u''(x) + q(x)u'(x) + r(x)u(x) = g(x)$$ In particular, note that any function dependent on $t$ multiplied by $u''(x)$ has been divided out, to ...
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1answer
23 views

Do we suppose that $y$ is the variable at which $f$ is differentiable in $\mathbb{R}$ ?

In my notes there is the following: The solution of the problem $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0 \\ u(x, 0)=f(x), x \in \mathbb{R} \\ u_t(x,0)=g(x)$$ is ...
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1answer
23 views

Canonical form of the differential equation

In my notes there is the following: Find the canonical form of the differential equation $$4u_{xx}-12u_{xy}+9u_{yy}+u_{y}=0$$ $$\Delta=(12)^2-4^2 \cdot 3^2=0$$ The canonical form will be of ...
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1answer
28 views

Difficult exercise on unicity of solutions for an IVP

Suppose $f$ and $g$ are continuous and $g$ is odd and strictly increasing function. I have to prove that the IVP $$y'=f(x)g(y)$$ $$y(0)=1$$ has a unique solution if and only if $$\lim \limits_{u \to ...
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2answers
35 views

Find $x(t)$ and $y(t)$ which satisfy the following differential equations

Find $x(t)$ and $y(t)$ which satisfy $3\dot x + \dot y +5x-y=2e^{-t}+4e^{-3t}$, $\dot x + 4\dot y -2x+7y=-3e^{-t}+5e^{-3t}$, subject to $x=y=0$ at $t=0$. This is how I tried it: If we multiply ...
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0answers
19 views

Eigenvectors question

$x'=x-2y$ $y'=4x-x^3$ Consider equilibrium point $(2,1)$: Let $X=x-2$ and $Y=y-1$. Subbing this into the main and eliminating all the nonlinear terms gives: $x'=X-2Y$ $y'=-8X$ Giving the ...
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1answer
34 views

Differential Equations: Linear or Nonlinear

In my textbook, the authors said that a differential equation is linear if it can be expressed in the form $$a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_n(t)y=g(t)$$ According to the definition, why the ...
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1answer
27 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
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2answers
22 views

What theorem can I use to decide if an ODE which admits separation of variables has a unique solution?

Suppose that I have the IVP : $$y' = f(x)g(y)$$ $$y(a)=b$$ It's easy to show that any solution of such an equation will satisfy the implicit formula: $$ \int \frac{1}{g(y)}dy = \int f(x) dx$$ I ...
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2answers
27 views

Finding an ODE given some of its solutions

Find $a, b, f(x)$ such that $$y''+ay'+by = f(x)$$ Is satisfied by $g_{1}=\sin x + e^x$ and $g_{2}=\sin x - e^{-x}$ What I tried to do: First, I used the fact that if $g_{1}$ and $g_{2}$ are ...
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1answer
44 views

Coefficient calculation on Fourier series !?

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
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1answer
51 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
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1answer
35 views

What does it mean for a difference equation to be made dimensionless?

Given the logistic difference equation, what does it mean for it to be made "dimensionless" so that it's written in a simpler form? $${N_{n+1}-N_n\over \Delta t}=r\cdot N_n\cdot \left(1-{N_n\over ...
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1answer
20 views

Find A,B such that the given function is a solution to the given differential equation

The equation is: $x(t)=A\sin(t)+B\cos(t), \; x' - 3x = \frac{1}{2}\cos(t)$ I'm honestly just lost from where to start. I'd really appreciate any help. Answer: $A = \frac{1}{20}, \; B= ...
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25 views

Getting solutions for an ODE in different intervals

I have to obtain solutions for: $$ x(x-1)y' + y = x^2 -1$$ In the intervals $(- \infty, 0), (0,1), (1,+ \infty)$. I got the solution $$f(x) = \frac {x}{x-1}(C+x+\frac{1}{x})$$, where $f$ is defined ...
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1answer
56 views

Solution of differential equation by conversion to an integral equation [on hold]

Let us say we have the following initial value problem : $$ y''+g(t,y)=0$$ such that $y(0)=y_0$ And $y'(0)=z_0$ . And $g$ is a continuous function in the domain $D$ contains the point $(0,y_0)$. ...
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1answer
45 views

How can I solve the following differential equation [on hold]

How can I solve the following differential equation : $$w''+(\sin z)w'+(1+z^2)w=0$$ In two case : without use power series use power series near the point $z=0$
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0answers
26 views

Prove that the solution of an ODE can be prolonged to $\infty$

I need an help understanding some general techniques in ordinary differential equations. I've never attended a course on ODE, so I'm quite confused on the argument, but I'm trying to improve my ...
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0answers
13 views

Limit Cycle of a Certain Autonomous Almost Linear D.E.

My task is to describe the stability of the system about the origin: $\frac{dx}{dt}=-2x+y+2y^2$ $\frac{dy}{dt}=-x+xy$ Im done with the stability part, but I am stack on how to determine if it has a ...
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1answer
40 views

Quick question that I can't find anywhere online about Runge-Kutta

I'm writing a presentation on modelling fluid flow. We used Runge-Kutta second order to describe the flow as a numerical method. I just want verify that Runge-Kutta fourth order would be of a higher ...
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1answer
19 views

Finding a solution for $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$).

How do I find a solution for the differential equation: $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$) by using the property of inverse function: $\frac{dx}{dy} = ...
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0answers
12 views

How do you determine if its an improper node of a proper node?

This is my example but your welcome to elaborate if you have better examples. Suppose we have a system of differential equations $$\frac{dR}{dt}=aJ \\ \frac{dJ}{dt}=bR$$ s.t. $a,b \gt 0$. I've ...
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1answer
24 views

Find a solution of the Laplace equation $-\Delta u=1$ with boundary condition $u=0$ on a spherical shell

Let $n\ge 2$ $B_\varepsilon$ and $\overline{B}_\varepsilon$ be the open and closed ball around $0$ with radius $\varepsilon>0$ in $\mathbb{R}^n$, respectively $R>0$, $\rho\in (0,R)$ and ...
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1answer
32 views

Flow property of a differential equation

In order to solve this question I must fix $s$ while allowing $t$ to vary. I am confused as to why this is allowed.
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24 views

for the following equation, state whether it is homogenous or inhomogeneous

$$y^{\prime\prime\prime}+y=9$$ I cant understand why this equation is inhomogenous. Homogenous is of the form $\dfrac{\mathrm dy}{\mathrm dx}+yP(x)=0$. Inhomogenous is of the form $\dfrac{\mathrm ...
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0answers
21 views

What would it mean if the solution to an IVP did not have maximal interval of existence?

I understand that it can be proved that for the IVP: $\frac{dx}{dt}=f(x)\quad$; $x(0)=x_0$ There is a maximal interval of existence $(\alpha,\beta)$, where there exists a unique solution. However I ...
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1answer
27 views

The time interval a solution exists for

I understand Q1, however it is required for Q2 Let $J(x_0)$ be the union of all open intervals $J(x_0)∋0$ st. $x_{Ji}$ is a solution of $x′=f(x),x(0)=x_0$ (∗) so $J(x_0)$ takes the form ...
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4 views

Deriving expression for the 'curve-in' condition in an equiangular (log/bernoulli spiral)

For any 2 pairs of points (xe,ye) & (xs,ys), I can fit various equiangular spiral through those 2 points based on the equation r = ke^(aθ). A typical one is illustrated below: Then, I can vary ...
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26 views

Separable ODE and singular solutions

In most introductory ODE textbooks we can find the following: A separable first-order ODE is the one of the form $$y'=g(x)h(y)$$ and if $h(y)\neq0$, then the general solution is found by integration ...
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0answers
21 views

What is a bifurcation point?

Given a density dependent difference equation, $N_{n+1}=N_{n}e^{r[1-(N_{n}/K)]}=f(N_{n})$, with $r > 0$ and $K > 0$. I've found that the equilibria are at $N^*=K$ or $0$. Discussing their linear ...
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Discrete logistic model

Given a difference equation such as N_{n+1}=N_{n}exp^{r[1-(N_{n}/K]=f(N_{n}). What does it mean when they say density-dependent difference equation?
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Given that e^t is a solution of the differential equation: (t-1)y''-ty'+y=0 find a fundamental set of solutions.

I'm not sure how to proceed. I tried variation of parameters, but I end up with this funky answer: v=c \int e^((t^2)/2+2t)
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1answer
15 views

Linear Differential Equation (Initial condition problem).

$$y'+ y= \frac{7\sin(t) }{ \sqrt{t + 1}}$$ Find the solution to this differential equation that satisfies the initial condition $y(0) = 1$. Your answer should be expressed in terms of a definite ...
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1answer
27 views

y''+y=cos(t) what is the smallest possible value of t for which |y(t)|>10?

Not sure if this is correct, but I was able to find a general solution of the form: y= c1cos(t)+c2sin(t)+(1/2)tsin(t) I'm not sure how I would go about finding the smallest possible value to make the ...
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Is this differential equation separable?

$$x\frac{dy}{dx}-y^2 = \frac{dy}{dx}+5$$ I have found that this equation is differentiable as shown in the following. $$x\frac{dy}{dx}-\frac{dy}{dx} = y^2+5$$ $$dy(\frac{x}{dx}-\frac{1}{dx}) = ...
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1answer
24 views

Sketching phase portrait

$\dot{x}=-2x-2y$ $\dot{y}=-x-3y$ Equilibrium point is $(0,0)$. Eigenvalues are $\lambda_+=-1$ and $\lambda_-=-4$ which have corresponding eigenvectors $2\choose -1$ and $1 \choose 1$ respectively. ...
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0answers
18 views

How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
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1answer
23 views

How to guess a change of variable for an ODE?

My question is related about how to guess what change of variable to make in an ODE? For example, we have $$\varepsilon^2y(x)''=axy(x)$$ where $\varepsilon\ll1$, $a$ is constant and $x$ is the ...
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0answers
11 views

Solving System of Boundary Value problem

The boundary value problem: $$y'' + Q(t)y = f(t)$$ satisfying $$Ay(a) +By(b) = g$$ where A, B and Q are the matrices of order n. After calculation, we can get the form of solution will be $$y(x) = ...
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0answers
16 views

Solving IVP using Laplace Transform

Let $$g(t) =\begin{cases} t & \text{if $t \leq6π$} \\ 6\pi & \text{if $t>6\pi$} \end{cases} $$ Solve $y''+ 16y = g(t)$ where $y(0) = 9$ and $y'(0) = 4$ using Laplace transforms. I got ...
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19 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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1answer
59 views

Solving $y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}$

There is the trivial $y=0$, but beyond that, could there be further solutions for $y$ in terms of $x$ such that $$y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}\mbox{ pointwise}$$ ? I posed this problem ...
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0answers
26 views

Inverse laplace transform of $1/(s^2 +1)^{1/2}$

I tried this problem using the basic properties but I am unable to get further than reducing this one to $(s^2+1)^{1/2}$.
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1answer
15 views

Lipschitz condition in non-autonomous case

I need to show the RHS of $\frac{dy}{dt}=-y^2+y+2yt^2+2t-t^2-t^4=f(t,y)$ is locally lipschitz. Am I able to use continuously differentiable implies locally lipschitz here with non-autonomous $f$? ...
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1answer
24 views

Series solution of a differential equation with a function as a coefficient?

I'm trying to solve a differential equation using series: the equation is $$y''+(\sin(x))y=0$$ I know you have to use the power series form of $\sin(x)$ and multiply it with the power series form of ...
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1answer
34 views

Black-Scholes Problem

Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 \hspace{3mm}\textrm{and}\hspace{3mm} ...
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0answers
26 views

Meaning of maximal interval of existence [on hold]

Does this mean the entire domain for which the solution of the IVP is continuous on before it blows up?