📘 differential equations

1. **State the problem:** Solve the differential equation $$9y^{(4)} - 12y'' + 4y = 0$$ with initial conditions $$y(0) = 1, y'(0) = 2, y''(0) = 3, y'''(0) = 4.$$\n\n2. **Form the c

1. **State the problem:** Solve the differential equation $$y'' + 6y' + 9y = 0$$ with initial conditions $$y(0) = 2$$ and $$y'(0) = -10$$. 2. **Find the characteristic equation:**

1. Problem a) Solve the Bernoulli differential equation $$\frac{dy}{dx} - x y = e^{-x^2} y^3$$ Step 1: Identify the Bernoulli form: $$\frac{dy}{dx} + P(x) y = Q(x) y^n$$ with $$P(x

1. **State the problem:** Solve the differential equation $$y(1 - xy^2)\,dx + dy = 0.$$\n\n2. **Rewrite the equation:** Express it in the form $$M(x,y)\,dx + N(x,y)\,dy = 0$$ where

1. **State the problem:** Solve the differential equation $ (2y + \sqrt{y} \sin x) \, dx + x \, dy = 0 $. 2. **Rewrite the equation:** The given equation is $ (2y + \sqrt{y} \sin x

1. **State the problem:** Solve the differential equation $$(\tan x - \sin x \sin y)\,dx + (\cos x \cos y)\,dy = 0.$$\n\n2. **Rewrite the equation:** We have $M(x,y) = \tan x - \si

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