📘 differential equations

1. **Stating the problem:** We are given the differential equation $$(X - y^2)dx + 2xy dy = 0$$ and need to check if it is an exact differential equation. 2. **Recall the definitio

1. **Stating the problem:** We are given a system of three linear differential equations:

1. **Stating the problem:** We are given a system of linear differential equations: $$\begin{cases} x' = 2x + y + 2z \\ y' = x + 2y + 2z \\ z' = x + y + 3z \end{cases}$$

1. **Problem:** Solve the first-order ODE $$\frac{dy}{dx} = -\frac{6x^2 y^2 + y^5}{6x^3 y^2 + 5xy^4}$$

1. **Problem:** Solve the first-order ODE $$\frac{dy}{dx} = -\frac{6x^2 y^2 + y^5}{6x^3 y^2 + 5xy^4}$$

1. **State the problem:** Solve the differential equation $$(D^3 - D^2 + D - 1)y = 4 \sin x,$$ where $D$ denotes differentiation with respect to $x$, i.e., $D = \frac{d}{dx}$. 2. *

1. **State the problem:** Solve the differential equation $$(D^3 + D^2 - 4D - 4)y = 3e^{-x} - 4x - 6$$ where $D$ denotes differentiation with respect to $x$. 2. **Rewrite the opera

1. **State the problem:** Solve the differential equation $$x^2 y'' + 3x y' + 5y = 8x$$ with initial conditions $$y(1) = 2$$ and $$y(e^{\pi/4}) = 2 \sinh(\pi/4)$$. 2. **Identify th

1. **Διατύπωση του προβλήματος:** Δίνεται η διαφορική εξίσωση $$\left(m+\frac{2I}{r^2}\right)s'' = \frac{1}{r}(t_r - t_l) - b s' s'$$ με αρχικές συνθήκες $$s(0) = s_0$$ και $$s'(0)

1. **State the problem:** Solve the differential equation $$(6x - 3y + 2)dx + (2x - y - 1)dy = 0.$$ 2. **Identify the type of equation:** This is a first-order differential equatio

1. **State the problem:** Solve the differential equation $$(3x^2 y - x^2) \, dx + dy = 0.$$ 2. **Rewrite the equation:** Express it in the form $$M(x,y) \, dx + N(x,y) \, dy = 0,$

1. The problem is to solve higher-order linear homogeneous ordinary differential equations (ODEs) by finding characteristic roots. 2. The general form of such an ODE is $$a_n y^{(n

1. **State the problem:** We have a linear system defined by the differential equation $$ay'' + by' + cy = f(t)$$ with initial conditions $$y(0) = 0$$ and $$y'(0) = 0$$.

1. **State the problem:** Solve the differential equation $$y'' - 7y' + 10y = -t^2 e^{2t}$$ with initial conditions $$y(0) = 3$$ and $$y'(0) = 9$$. 2. **General solution form:** Th

1. **State the problem:** We want to find the Laplace transform $Y$ of the solution $y(t)$ to the initial value problem (IVP):

1. **State the problem:** Solve the initial value problem using the Laplace transform: $$y'' + 3y' = 0, \quad y(0) = 6, \quad y'(0) = 4$$

1. **Problem statement:** Solve the ordinary differential equation (ODE) $$y' = y - y^3$$ with the initial condition $$y(0) = R > 0$$. 2. **Identify the type of ODE:** This is a se

1. **Problem statement:** Solve the ordinary differential equation (ODE) $$y' = y - y^3$$ with the initial condition $$y(0) = R$$ where $$R > 0$$. 2. **Rewrite the ODE:** The equat

1. **State the problem:** Solve the ordinary differential equation (ODE) given by $$y' = y - y^3$$. 2. **Identify the type of ODE:** This is a separable differential equation becau

1. The problem asks: Which of the following represents the general standard form of a first-order linear differential equation? 2. The standard form of a first-order linear differe

1. The problem asks: Which of the following represents the general standard form of a first-order linear differential equation? 2. The general standard form of a first-order linear