📘 differential equations

1. **State the problem:** Solve the differential equation given by $$y\,dx - x\,dy = xy\,dx.$$\n\n2. **Rewrite the equation:** Move all terms to one side to better analyze it:\n$$y

1. The problem is to define an exact differential equation and provide an example. 2. An exact differential equation is a first-order differential equation of the form $$M(x,y) + N

1. **State the problem:** Find the general solution of the differential equation given implicitly by $$y = px + \frac{a}{p}$$ where $p = \frac{dy}{dx}$. 2. **Rewrite the equation:*

1. The problem is to solve the first-order linear differential equation $$\frac{dy}{dx} + 3xy = \sin x.$$\n\n2. We are given the integrating factor $$u = e^{\frac{3}{2} x^{2}}.$$\n

1. The problem is to find the Laplace transform of the sum of two functions, say $f(t)$ and $g(t)$. 2. Recall the linearity property of the Laplace transform: $$\mathcal{L}\{f(t) +

1) Solve the differential equation $\left(y + \sqrt{x^2 - y^2}\right) dx + x dy = 0$. Step 1. Rewrite the equation:

1. **Problem 1:** Solve the differential equation $\left(y + \sqrt{x^2 - y^2}\right) dx + x dy = 0$. Step 1: Rewrite the equation as $\left(y + \sqrt{x^2 - y^2}\right) + x \frac{dy

1. Let's start by stating the problem: solve the differential equation $$\frac{dy}{dx} = 2x^2 + y - x^2 y + x y - 2x - 2.$$\n\n2. First, rewrite the right side to group terms invol

1. Stating the problem: We are given the differential equation $$\frac{dy}{dx} = 2x^2 + y - x^2 y + x y - 2x - 2$$ and we want to analyze or solve it. 2. Simplify the right-hand si

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = 2x^2 + y - x^2 y + x y - 2x - 2$$ using the separable variable method. 2. **Rewrite the equation:** Grou

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} \ln x + y = e^x$$ for $y$ as a function of $x$.\n\n2. **Rewrite the equation:** The equation is $$\frac{dy

1. **State the problem:** Find the differential equation for the function $$y = A \sin 3x + B \cos 3x$$ where $A$ and $B$ are constants. 2. **Differentiate the function:** Compute

1. **State the problem:** We have a function $f(x)$ satisfying the differential equation $$f''(x) = 6x f'(x) + (4x^2 - 2)f(x)$$

1. **State the problem:** We have a function $f(x)$ satisfying the differential equation $$f''(x) = 6x f'(x) + (4x^2 - 2)f(x)$$

1. **State the problem:** We have a function $f(x)$ satisfying the differential equation $$f''(x) = 6x f'(x) + (4x^2 - 2)f(x)$$

1. **State the problem:** We have a function $f(x)$ satisfying the differential equation $$f''(x) = 6x f'(x) + (4x^2 - 2) f(x)$$ with initial conditions $$f(0) = 1, \quad f'(0) = 0

1. The problem asks whether the differential equation $$\frac{dy}{dx} = y + 3$$ is separable. 2. A differential equation is separable if it can be written in the form $$\frac{dy}{d

1. **State the problem:** Simplify and solve the differential equation $$3(y + 2) \, dx - x y \, dy = 0$$. 2. **Rewrite the equation:** We have $$3(y + 2) \, dx = x y \, dy$$ or eq

1. **State the problem:** We need to simplify and solve the expression $$Y_p = -e^x \int \frac{e^{-x}}{-2} x^2 \, dx + e^{-x} \int \frac{e^x}{-2} x^2 \, dx$$

1. The problem is to express the differential equation $$y''' - y' + y = \cos t$$ in operator form. 2. Define the differential operator $$D = \frac{d}{dt}$$.

1. **State the problem:** Solve the differential equation $$y'' - 9y = -24e^{-2t}$$ with initial conditions $$y(0) = 8$$ and $$y'(0) = 0$$. 2. **Solve the homogeneous equation:** T