# Tagged Questions

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

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### Differential Equations on Population

I've been beginning to try to solve some simple differential equations. I reached a problem where there is a polar bear population $p(t)$ and a seal population $s(t)$ that are governed by ...
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### Linearization Theorem. Graphs from initial conditions and a DE

This is a problem I have. I am given a DE $(\star)=(y-1)sin(y)$ and told to sketch the phase lines for $(\star)$ and identify whether they are sinks, sources, or nodes. That part I was able to do. ...
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### Sturm Liouville problem for $x^2φ''+ xφ' + λφ = 0$

Consider the eigenvalue problem $$x^2φ''+ xφ' + λφ = 0, \quad 1<x<2, \quad φ(1) = 0, φ(2) = 0$$ (a) Write the problem in Sturm-Liouvile form, identifying $p, q,$ and $σ$. (b) Is the problem ...
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### Show $f_t- a x f_x + \frac{1}{2} b \sigma^2 f_{xx}=0$ has solutions of the form $f(x,t) = C(t) e^{-D(t)x^2}$

Show $$\partial_t f(x,t) - a x \partial_x f(x,t) + \frac{1}{2} b \partial_x^2 f(x,t)=0$$ has solutions of the form $$f(x,t) = C(t) e^{-D(t)x^2}$$ and find the differential equations ...
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### need to understand better ODE solution accuracy vs numerical precision

NOTE: I first posted this in python SO but possibly this site is better suited. I need to understand the relationship betwen ode solution accuracy and numerical precision. Here ...
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### Check if there is a solution of the differential equation

In my notes there is the following: $$y''+y'=1$$ The particular solution is a polynomial, say $f$. The solution of the homogeneous equation is $y_H=c_1+c_2e^{-x}$. Therefore, the solution of ...
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### Reducing a 2nd order system of ODEs to a 1st order system

I need to numerically solve the following system of ODEs: $$x''(t)=- \frac{3x}{(x^2+y^2)^{3/2}}$$ $$y''(t)=- \frac{3y}{(x^2+y^2)^{3/2}}$$ I know that I must convert to a system of 1st order ...
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### Determining the size of a basin of attraction

$$\dot x = -2x-y^2$$ $$\dot y = -y-x^2$$ $(0,0)$ is an obvious attractive fixed point, and I'll only look at this one. I need to get the maximal radius $r > 0$ for a ball centered on the origin ...
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### I am having trouble figuring out how to get the characteristic equation for this symmetric matrix to then help me get the eigenvalues. [closed]

I am new to linear algebra and I am having trouble figuring out how to get the characteristic equation for this symmetric matrix. $$A=\left(\begin{matrix}a&b\\ b&a\\ \end{matrix}\right)$$ ...
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### How to calculate step response for $y''(t) - y(t) = x'(t) - x(t)$ in time domain?

How to calculate step response for $y''(t) - y(t) = x'(t) - x(t)$ in time domain? So, without Laplace or Fourier Transforms. This is what I tried: The Homogeneous solution of the differential ...
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### solving analytical solution using matlab

I have the solution for a second order differential equation as follows and for varying x, I want to find the temperature. with the parameter values being, and...
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### Differential Equation problem

Find $y$, when $xy'-x=1$ and $y(1)=2$. I get that $y=x+\ln{ x}+C$ But then $y(1)=2$ isn't true. Did I do something wrong?
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### Describe all differentiable functions that follow these rules

Describe all differentiable functions that follow these rules: $$f'(x) = f(x)^3$$ and $$f(0) = 2.$$ This came up in the scholarship exam today. I am clueless as to how you do this one.
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### prove that the general solution of a homogenous Second order DE is unique or any other solution of the DE is in the form of the general solution?

Let $y_1,y_2$ be fundamental set of solutions of a homogenous second order DE, I want to show that any other solutions came from or is a combination of $y_1$ and $y_2$ and is a general solution ...
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### How to perform taylor expansion with numerical differentiation formula

I am attempting to perform taylor expansion on the following numerical differentiation formula: $f'''(0) = \frac {−f(−3h/2) + 3f(−h/2) − 3f(h/2) + f(3h/2)) }{ h^3 }$ Over the reference interval ...
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### solve differential equation $f'(x)=f(x)$

I want to solve the differential equation $f'(x)=f(x)$ using power series of the form $$f(x)=\sum_{n=0}^{\infty}{c_nx^n}$$ From my previous knowledge I know that the solution is $f(x)=c_0e^x$ I can ...
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### Can a power series solution be found for this? y''-y'-6y=5e^7x.

My teacher wants us to use power series and not the auxiliary equation, but I can't get this to fold into a single series. Original Question: "Using power series, find the general solution of ...
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### Finding the particular solution for $y'' - y' = -3$

I am trying to find the general form of the particular solution for this problem. I originally tried $y_p = A$ but that did not give me the correct answer. Do I have to do something different if the ...
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### Integrating factor for differential equations

okay so I think I did it correctly but the answer looks rather odd. Question: Find the general solution of the DE Reasoning: $\frac{dy}{dt} + 2ty = 4e^{-t^{2}}$ $\mu(t)=e^{t^{2}}$ ...
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### Solving an exact differential equation (using an integrating factor)

Given this differential equation $x^2y^3+y+(x^3y^2-x)y'=0$ I have to find an integrating factor, such that the equation becomes an exact differential equation. I am pretty sure that the integrating ...
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### Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? [duplicate]

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? In particular I'm interested in the qualitative properties of the such solutions.
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### solve the following first order but not of first degree ordinary differential equation

$$x = y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^2$$ I think it can be converted into a Clairaut. form but not able to do it
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### Proof for D'alembert's solution

I am totally clueless in how to go about the solution for part iii). I was able to answer the others. Normally I would find the derivatives then substitute. In regards to the initial conditions I'm ...
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### Finding an integration factor $\mu(xy)$ for first order ODE for a nonexact equation.

I've been searching the internet for how to find an integration factor for first order ODE, and I found something, but I didn't quite get the steps for getting what they got. I added a screenshot ...
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### Maximal domain of existence for non-autonomous system

Using Picard-Lindelof Existence and Uniqueness Theorem, I need to find the maximal domain of existence for $x′=tx^3, x(0)=x_0$. I do not know how to do it for a non-autonomous system. If there was no ...
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### How to reduce the chemical reaction system $A+B \rightleftharpoons^{k_+}_{k_-} C$ with transfer using the quasi steady state assumption

Suppose we have the reaction $$A+B \rightleftharpoons^{k_+}_{k_-} C$$ within some reaction $\Omega\subset\mathbb{R}^3$, we assume this region to be well mixed and we denote the concentration of $A$ as ...
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### How to prove $X(t)$ is differentiable?

Suppose $X(t)\in M_n(\Bbb R), t\in\Bbb R$ and is continuous, invertible at every point on the real line, if the equation $$X(t)X(s)=X(t+s)$$ holds for all $t,s\in\Bbb R$, prove that there exists a ...
Solve $\dfrac{dy}{dx}=\dfrac{y-3}{y^2+x^2}$ given that it passes through $(0,1)$. Right now I do not yet know how to solve differential equations with both $x$ and $y$ that you cannot separate. ...
Can anyone help me in finding solution of the LTV equation given by $$t\ddot{x}-\dot{x}+tx=0$$ or equivalently the LTV system $$\dot{x}_1 =-tx_2, \dot{x}_2 =-\frac{x_1}{t}$$. Thanks for your time ...