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### Need explanation for simple differential equation

I can't figure out this really simple linear equation: $$x'=x$$ I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating ...
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I've read journal "On the Comparison of Several Mean Values: An Alternative approach" (Welch, 1951). I don't understand this expression: $$E\left(\exp\left[ \sum_t ( w_t - \omega_t ) ... 3answers 220 views ### What are other solutions to this differential equation, “similar” to \sin x and e^x? I've been studying electronics, where they make great use of the relationship between the sine and exponential functions (e^{i \omega t} = \cos{\omega t} + i \sin \omega t). This relationship is ... 0answers 184 views ### maple code for exp-func. for solving PDE's & non-linear ODE's? How can I create the Maple code using exponential-function solving the equation below? u_t = \gamma u_x+6u(u_x)^2+(3u^2-1)u_{xx}-u_{xxxx} u_t =u_{xx}-u^3+u, \alpha u''(x) = \beta ... 1answer 217 views ### Need to deduce f(x) from f_x=e^{t(x)} I know that$$f_x=e^{t(x)}$$(where the notation f_x=\frac{df}{dx}) (EDIT: f=f(x) and t parameterizes x, so x=x(t) \Leftrightarrow t=t(x)) and that therefore$$\frac{d^n ...
I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$ I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$ at ...