📘 differential equations

1. **State the problem:** Solve the differential equation $$\sqrt{1-x^2} \, dy + \sqrt{1-y^2} \, dx = 0$$ using the separable method and find the explicit solution. 2. **Rewrite th

1. **State the problem:** We are given the function $$y = C_1 x + C_2 x^2$$ with arbitrary constants $$C_1$$ and $$C_2$$.

1. **Problem (a):** Solve the initial value problem $$y'' + 3y' - 4y = 6e^{2x} - 3$$ with initial conditions $$y(1.5) = 4$$ and $$y'(1.5) = 5$$. 2. **Step 1: Solve the homogeneous

1. The problem asks to find the equilibrium points from a given image. 2. Equilibrium points occur where the system's derivatives are zero or where the function does not change.

1. **Problem Statement:** We address multiple problems involving second order ordinary differential equations (ODEs) and Laplace transforms.

1. **Problem Statement:** We have a kitchen of volume $60$ cubic metres. Smoke is produced at a rate of $0.06$ mg/s and the air conditioning exchanges air at $6$ cubic metres per m

1. **State the problem:** We need to solve the differential equation $$y' = y$$ where $y'$ denotes the derivative of $y$ with respect to $x$. 2. **Recall the formula:** This is a f

1. **State the problem:** Solve the initial value problem $$y'' - 6y' + 5y = 3e^{2t}$$ with initial conditions $$y(0) = 2$$ and $$y'(0) = 3$$ using the Laplace transform method. 2.

1. **State the problem:** Solve the initial value problem $$y'' - 6y' + 5y = 3e^{2t}$$ with initial conditions $$y(0) = 2$$ and $$y'(0) = 3$$ using the Laplace transform method. 2.

1. **Problem statement:** Solve the initial value problem (IVP) by inverting the Laplace transform for the differential equation $$y'' - 3y' - 10y = 1$$ with initial conditions $$y

1. **State the problem:** Solve the partial differential equation (PDE) given by $$\frac{dy}{dx} = \frac{3y^2 - x^2}{2xy}$$. 2. **Rewrite the equation:** The equation can be writte

1. **State the problem:** Solve the first-order linear differential equation $$\frac{dy}{dx} + \frac{y}{x} = \sin x, \quad x > 0.$$ Find the general solution for $y$ in terms of $x

1. Das Problem lautet: Finde die allgemeine Lösung des Systems $$\begin{cases} x' = \alpha x + \beta y \\ y' = -\beta x + \alpha y \end{cases}$$

1. Das Problem: Wir suchen ein Beispiel für eine Differentialgleichung $y' = f(t,y)$, bei der der Peano-Existenzsatz gilt, aber der Satz von Picard-Lindelöf nicht anwendbar ist. 2.

1. Das Problem: Wir betrachten Anfangswertprobleme (AWP) gewöhnlicher Differentialgleichungen (DGL) der Form $$y'(t) = f(t,y(t)), \quad y(t_0) = y_0$$. 2. Existenz- und Eindeutigke

1. **Problem:** Solve the differential equation $$x\,dx + y e^{-x} dy = 0$$ with initial condition $$y(0) = 1$$. 2. **Formula and rules:** This is a separable differential equation

1. **State the problem:** Solve the differential equation $$y''' - 3y'' + 3y' + y = 4e^t$$ with initial conditions $$y(0) = 1$$, $$y'(0) = 1$$, and $$y''(0) = -1$$. 2. **Characteri

1. **State the problem:** Solve the differential equation $$y''' - 3y'' + 3y' - y = 4e^t$$ with initial conditions $$y(0) = 1$$, $$y'(0) = 1$$, and $$y''(0) = -1$$. 2. **Identify t

1. **Stating the problem:** We want to solve the differential equation $$\frac{dy}{dx} - \frac{y}{x} = 2x$$ where $y$ is a function of $x$. 2. **Identify the standard form:** The e

1. **Stating the problem:** We want to solve the differential equation $$\frac{dy}{dx} - \frac{y}{x} = 2x$$ where $x \neq 0$. 2. **Identify the standard form:** The equation is lin

1. **State the problem:** Solve the differential equation $$\left(1-\frac{3}{y}+x\right)\frac{dy}{dx}+y=\frac{3}{x}-1$$ for $y$ as a function of $x$.