📘 differential equations

1. **State the problem:** Solve the differential equation $$y^{(5)}+3y^{(4)}+y^{(3)}-y^{\prime\prime}-4y=0$$. 2. **Characteristic equation:** For linear differential equations with

1. The problem is to understand how to plot a phase portrait using GeoGebra. 2. A phase portrait is a graphical representation of a dynamical system's trajectories in the phase pla

1. **State the problem:** We want to solve the differential equation $$y' = ky$$ with $$k=1$$ using a power series approach. We assume the solution has the form $$y = f(x) = \sum_{

1. **Problem statement:** We have the autonomous system of differential equations: $$\begin{cases} x' = -x + xy \cos(x), \\ y' = -y + yz \sin(y), \\ z' = -z, \end{cases}$$

1. **Stating the problem:** We have the system of ordinary differential equations (ODEs): $$X' = X(1 - Y)$$

1. **Problem Statement:** We want to explore a logistic differential equation and understand its bifurcation diagram. 2. **Logistic Differential Equation:** The logistic equation m

1. The problem is to provide an example of a logistic differential equation. 2. A logistic differential equation models population growth with a carrying capacity, limiting the gro

1. **Énoncé du problème :** Nous avons une équation différentielle (E1) : $$y' = 0,022y(20 - y)$$ où $f(t)$ représente le nombre de ménages équipés d'un ordinateur en millions, $t$

1. **State the problem:** We need to find the general solution of the system of differential equations: $$\frac{dx}{dt} = 3x - y$$

1. **State the problem:** Solve the differential equation $$y' + \frac{2y - 1}{x} = x + 1$$. 2. **Rewrite the equation:** Distribute the fraction:

1. **State the problem:** Solve the first-order linear differential equation $$\frac{dy}{dx} + (\tan x) y = \sin 2x$$ with the initial condition $$y(0) = 1$$. 2. **Identify the sta

1. **State the problem:** Solve the differential equation $$\frac{dB}{dt} + 0.3B = 0.02B^2$$ with initial condition $$B(0) = 1$$. 2. **Identify the type of equation:** This is a Be

1. **State the problem:** Solve the differential equation $$y''' - 2y'' - y' + 2y = 4e^{3x} + 6x$$. 2. **Find the complementary solution (homogeneous equation):** Solve $$y''' - 2y

1. **Planteamiento del problema:** Se tiene la ecuación diferencial $$\frac{dB}{dt} + 0.3B = 0.02 B^2$$ con condición inicial $$B(0) = 1$$. Se busca encontrar la función $$B(t)$$ q

1. **State the problem:** Solve the differential equation $$y''(x)\cos(2x) + y'(x) = 0.$$ 2. **Rewrite the equation:** Let $y' = p$, then $y'' = p'$. The equation becomes $$p'(x)\c

1. **State the problem:** Solve the differential equation $$ (y^3 + y^2 x) \, dx - x^3 \, dy = 0 $$. 2. **Rewrite the equation:** We can write it as $$ (y^3 + y^2 x) + (-x^3) \frac

1. **State the problem:** Solve the differential equation $$ (2x - 5y + 1) \, dx + (-4y + 3x - 2) \, dy = 0 $$. 2. **Check if the equation is exact:** Let $$M = 2x - 5y + 1$$ and $

1. **State the problem:** Solve the differential equation $$(-4x + 5y + 8) \, dx + (6x - 9y + 4) \, dy = 0.$$\n\n2. **Check if the equation is exact:** Let \(M = -4x + 5y + 8\) and

1. **State the problem:** Solve the differential equation $$(-4x + 5y + 8) \, dx + (6x - 9y + 4) \, dy = 0.$$\n\n2. **Check if the equation is exact:** Let \(M = -4x + 5y + 8\) and

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{7 - 4x - 4y}{x + y - 3}$$ and we want to analyze or solve it. 2. **Identify the type of dif

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