📘 differential equations

1. **Problem Statement:** We consider the differential equation $\dot{x} = f(x)$ where $f(x)$ is given by the graph described. 2. **Phase Line and Equilibria:** Equilibrium points

1. Énonçons le problème : On doit résoudre une équation différentielle en utilisant la transformée de Laplace. 2. Rappel de la définition de la transformée de Laplace : Pour une fo

1. **Énoncé du problème :** Résoudre l'équation différentielle $$ (X + 1) \frac{d\sigma}{dX} - X \sigma + 1 = 0 $$ avec la condition initiale $$ \sigma(0) = 2 $$ en utilisant la tr

1. **State the problem:** Find the integral curves of the system given by the equation $$\frac{dx}{x+z} = \frac{dy}{y} = \frac{dz}{z+y^2}$$. 2. **Understand the problem:** Integral

1. **State the problem:** Find the differential equations of the space curves formed by the intersection of the two families of surfaces given by: $$u = x^2 + y^2 + z^2 = c_1$$

1. **State the problem:** Solve the initial value problem $$\frac{dy}{dx} = \frac{y^2 - \cos x \sin x}{x(1 - x^2)}$$ with initial condition $$y(0) = 2$$. 2. **Analyze the different

1. Problem: Solve the differential equation $xy \, dx + y^2 \, dy = 0$. 2. Formula and rules: This is a separable differential equation. We can rewrite it to separate variables $x$

1. The problem is to identify whether a given differential equation (DE) is an ordinary differential equation (ODE) or a partial differential equation (PDE), and to state its order

1. **Problem A1:** Solve the differential equation $$y'' - 5y' + 6y = 0.$$ 2. **Formula and rules:** This is a second-order linear homogeneous differential equation with constant c

1. **Problem Statement:** We have water flowing from an inverted conical tank with the rate of change of water height given by

1. **State the problem:** Solve the system of differential equations given by $$\mathbf{x}' = A\mathbf{x}$$ where $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ and $$A

1. The problem is to eliminate the arbitrary constants $A$ and $B$ from the function $$y = Ax^2 + Be^{2x}$$. 2. This function is a linear combination of two independent functions:

1. **State the problem:** We are given the function $$y = Ae^{2x} + Bxe^{2x}$$ where $A$ and $B$ are arbitrary constants.

1. **Problem:** Eliminate the arbitrary constant $C$ from the equation $$y = Cx + C^2 + 1$$ 2. **Step 1: Differentiate both sides with respect to $x$**

1. **Problem 1: Solve the differential equation** Given: $$y' = y - \frac{2x}{y}, \quad y(0) = 1$$

1. **State the problem:** A tank initially contains 100 gallons of fresh water. Saltwater with concentration $\frac{1}{2}$ lb/gal flows in at 2 gal/min, and the mixture flows out a

1. Diberikan persamaan diferensial koefisien linier nonhomogen: $$ (a_1 x + b_1 y + c_1) dx + (a_2 x + b_2 y + c_2) dy = 0 $$

1. **State the problem:** Solve the partial differential equation (PDE) $$x^2 r + 2 x y s + y^2 t = 0$$ where $r = \frac{\partial^2 z}{\partial x^2}$, $s = \frac{\partial^2 z}{\par

1. **Nyatakan masalah:** Selesaikan persamaan pembezaan linear tak seragam $$3y'' + 7y' + 2y = (x+1)e^{-2x}$$

1. **Problem statement:** Solve the second order differential equation $$y'' - 2y' + y = e^x$$. 2. **Find the complementary solution $g(x)$:**

1. **State the problem:** Solve the differential equation $$\sqrt{x+y+1} \frac{dy}{dx} = x + y - 1.$$\n\n2. **Rewrite the equation:** Let $z = x + y$. Then $y = z - x$ and $$\frac{