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Differential Equations on Math.Stackexchange
- how to solve this differential equation using the laplace transform?There was a physics problem and I got this differential equation $$\frac{d^2x}{dt^2}=g-\left(\frac{1}{x}\right)\left(\frac{dx}{dt}\right)^2$$ where $g$ is a constant and $t$ is time. I can easily solve this by using the product rule of derivatives, but my question is how to solve this equation using the Laplace transform. I do not know how to use the Laplace […]
- Is this differential equation linear?Assume we have (time dependent) quantities $a(t)$, $b(t)$ for which $$\frac{d}{dt} (a(t) b(t))=0$$ In simple words: The product of $a$ and $b$ is constant over t. The question: is this differential equation linear or non-linear? On the one hand it is a linear DG for $H(t)$, when the dependent variable is the product: $$H(t) = […]
- Why are the initial conditions in FEUT the way they are?The fundamental existence and uniqueness theorem for scalar linear ordinary differential equations states that for the initial value problem given by differential equation $ \sum_{i=0}^n a_{i}(x) y^{(i)} = f,$ where $y^{(i)}$ is the $i$th derivative of $y$, and the initial conditions $y^{(i)}(x_0) = c_i$ for $i = 0$ through $ n - 1$, there exists […]
- How should I apply the averaging theorem in this problem and show the following approximation?Consider the differential equation $\ddot{x}+\epsilon h(x, \dot{x})+x=0, 0
- Which kind of non-linear component is required for a differential equation in order to show frequency mixing?Which kind of non-linear component is required for a differential equation in order to show frequency mixing? Please be patient, maybe I am using mistaken terminology. If so, I ask you to correct me please. I going to focus the question on the electromagnetic wave equation as an example, by I am asking the question […]
- Power series solution using Leibniz-Maclaurin method of the Airy differential equationI'm trying to get a power series solution to the differential equation $ y'' - xy = 0, y(0) = 1, y'(0) = 2 $, using two different methods, expanding about $ x = 0 $. The first method is the 'normal' way of doing power series solutions, where we substitute $ y = \sum_{n=0}^{\infty} […]
- Solving an inhomogeneous ODE using Green’s functionsI'm trying to solve the following ODE using Green's function $$ y''(x) + 4y(x) = e^{x/2}$$ with the following boundary conditions: $$ y(0) = y(\pi / 4 ) = 0 $$ So I start by solving the homogenous part: $$ y_h'' + 4y_h = 0 \quad \rightarrow \quad y_h = Asin(2x) + B\cos(2x) $$ Applying […]
- Two answers to a differential equationI have the following differential equation: $x\mathrm dx + y\mathrm dy = x\mathrm dy - y\mathrm dx$ Method-1: Notice that, $x\mathrm dx + y\mathrm dy = \frac{\mathrm d(x^2 + y^2)}{2}$ so we substitute it for the LHS. After which multiplying and dividing RHS by $x^2$, Equation becomes: $\frac{\mathrm d(x^2 + y^2)}{2} = x^2\mathrm d\left(\frac{y}{x}\right)$, we […]
- How to solve this coupled differential equations? [closed]I don't have anything in my personal math toolbox that can help me to start finding $\eta(\tau)$ and $\xi(\tau)$ that satisfies this nasty looking set of differential equations: $$\frac{d^2\eta}{d\tau^2}+\frac{2}{\xi}\frac{d\xi}{d\tau}\frac{d\eta}{d\tau}=0$$ $$\frac{d^2\xi}{d\tau^2}+\xi\left(\frac{d\eta}{d\tau}\right)^2=0$$ Are there any ansatz I can try?
- How to calculate the flow of given vector field [closed]On the $\mathbb R^2$, there is vector field $$ V(x,y) = x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y} \tag{1} $$ Then, how to calculate the flow of $V$ ? The flow of $V$ is $\varphi_t: \mathbb R^2 \rightarrow \mathbb R^2$, where $t\in (-\varepsilon, \varepsilon)$, such that $$ \frac{d}{dt} \varphi_t(p) = V(\varphi_t(p)) \tag{2} $$ for any $t\in (-\varepsilon, […]
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