# Tagged Questions

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### existence and uniqueness of volterra integral equation of the first kind

$$\int_0^t k(s,t)f(s)ds=g(t)$$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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### Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
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### I need to solve $\dfrac{dx}{dt}= 2x(1-0.0001x)-0.01xy, \dfrac{dy}{dt} = -0.5y+0.0001xy$

I need to solve \begin{align} \frac{dx}{dt} &= 2\,x\,(1-0.0001\,x)-0.01\,x\,y \\ \frac{dy}{dt} &= -0.5\,y+0.0001\,x\,y \end{align} Can anyone tell how do we solve such problems, if ...
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### what can we say about the solution of the equation $y'=-y^2$ just by looking at it. [without finding its solution]

I trying to understand differential equations without finding their solution. This is a simple one, so I can verify the ideas from the solution. All ideas are appreciated, since they will help my ...
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### numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
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### Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
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### How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$-C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A$$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
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### Integral curves and ODE [closed]

How do I find a first-order differential equation having the given family ofcurves as integral curves to that one: all circles through the points $(1, 1)$ and $(-1,-1)$?
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### Integration and ODE

How do I integrate this? $$y'=1+\frac{y}{x}$$ I just don't know how to start. I think I gotta to try some variable changing, but I don't think I'm gonna so far with this.
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### Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...
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### Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
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### Integral with square root of function of function

I have the function $y=y(x)$ with $y'=dy/dx$, and the following equation: $ky'=\pm\sqrt{k^{2}-y^{2}}$, where $k$ is constant. Integrating this, given that $y(0)=0$, should give: $y=k\sin(x/k)$. I ...
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### On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y)$$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...