📘 differential equations

1. Solve the differential equation $$\frac{d^3y}{dx^3} + 3 \frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + y = e^{-x}$$ Step 1: Find the characteristic equation of the homogeneous part:

1. **State the problem:** We want to find which graph represents the solution $y(t)$ to the initial value problem (IVP): $$y'(t) = 4\left(y(t) - \frac{1}{4}y(t)^3\right), \quad y(0

1. The problem presents five differential equations for $\frac{dy}{dx}$: - А: $\frac{dy}{dx} = x + y$

1. The problem asks to find the differential equation that generates the given direction field. 2. A direction field shows the slope $\frac{dy}{dx}$ at each point $(x,y)$.

1. The problem asks to find the differential equation that generates the given direction field. 2. From the description, the vector field has a vertical component generally pointin

1. Задачата е да намерим наклона на късите отсечки в полето на направленията за диференциалното уравнение $$\frac{dy}{dx} = xy$$ в дадени точки. 2. За да намерим наклона в точка $$

1. **State the problem:** Solve the differential equation $$\frac{ds}{dt} + 2s = s t^2$$. 2. **Rewrite the equation:** Move all terms to one side:

1. Let's start by understanding what a differential equation is. A differential equation is an equation that involves an unknown function and its derivatives. It describes how a qu

1. The problem is to analyze the differential equation $$\frac{dN}{dt} = \frac{1}{10} N \left(3 - \frac{N}{3400}\right)$$ which models the rate of change of a population $N$ over t

1. The problem asks which graph represents the solution to the logistic differential equation $$\frac{dP}{dt} = P \cdot \left(2 - \frac{P}{10}\right).$$ 2. This is a logistic growt

1. Задачата е да намерим броя на хората $N(t)$, които са приели стила, когато броят им се увеличава най-бързо. 2. Диференциалното уравнение е $$\frac{dN}{dt} = N \left(0.1 - \frac{

1. The problem gives a differential equation describing the rate of change of $N(t)$: $$\frac{dN}{dt} = N \cdot \left(90000 - \frac{3N}{20000}\right)$$

1. **Problem statement:** We need to identify which graph represents the solution to the logistic differential equation $$\frac{dP}{dt} = 2P(50 - P)$$

1. Задачата е да намерим кривата на решенията на диференциалното уравнение $$\frac{dy}{dx} = \sqrt{xy}$$, която минава през точката $(0,9)$.\n\n2. Започваме с уравнението: $$\frac{

1. Consider the equation $$\sqrt{3 + \left(\frac{dy}{dx}\right)^2} = \sec x.$$\n\n(i) Order of the equation:\nThe order of a differential equation is the highest order derivative p

1. Consider the equation $$\sqrt{3 + \left(\frac{dy}{dx}\right)^2} = \sec x.$$\n\n(i) Order of the equation:\nThe order of a differential equation is the order of the highest deriv

1. Consider the equation $$\sqrt{3 + \left( \frac{dy}{dx} \right)^2} = \sec x.$$\n\n(i) Order of the equation:\nThe order of a differential equation is the highest order derivative

1. **State the problem:** Solve the differential equation $$d^4y/dx^4 + d^3y/dx^3 = 1 - e^{-x}$$. 2. **Rewrite the equation:** Let us denote derivatives as $$y^{(n)} = \frac{d^n y}

1. The problem is to solve the differential equation $$\frac{d^4 y}{dx^4} + \frac{d^3 y}{dx^3} = 1 - e^{-x}.$$\n\n2. First, write the equation as $$y^{(4)} + y^{(3)} = 1 - e^{-x}.$

1. The problem is to solve the differential equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx} = 1 - e^{-x}$$. 2. First, solve the homogeneous equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx

1. Stating the problem: Solve the differential equation $y\,dx + x\,dy = 0$. 2. Rewrite the equation in terms of $\frac{dy}{dx}$: