📘 differential equations

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

1. **State the problem:** Solve the differential equation $$y'' - 4y = 8e^{3t}$$ with initial conditions $$y(0) = 4$$ and $$y'(0) = 0$$ using Laplace transforms. 2. **Take the Lapl

1. **State the problem:** Solve the differential equation $$y'' - 4y = 8e^{3t}$$ with initial conditions $$y(0) = 4$$ and $$y'(0) = 0$$ using Laplace transforms. 2. **Take the Lapl

1. **State the problem:** Solve the system of differential equations with initial conditions: $$\frac{dy}{dt} = 2x + y$$

1. **State the problem:** Solve the differential equation $$x(1 + y^2)^{1/2} \, dx = y(1 + x^2)^{1/2} \, dy$$ and find the explicit solution. 2. **Rewrite the equation:** We can wr

1. **Problem 11:** Solve the differential equation $$\csc y \, dx + \sec^2 x \, dy = 0$$ with separation of variables. 2. Rewrite the equation as $$\csc y \, dx = -\sec^2 x \, dy$$

1. **State the problem:** Solve the differential equation $$\csc y \, dx + \sec^2 x \, dy = 0$$ by separation of variables. 2. **Rewrite the equation:** We have

1. **State the problem:** Find the general solution to the differential equation $$y'' - 2y' + y = \frac{1}{6} e^{5x} \cos(3x).$$ 2. **Solve the homogeneous equation:** The associa

1. The problem is to solve the differential equation for $x$. 2. Since the exact differential equation is not provided, let's consider a general first-order differential equation o

1. The integrating factor is used to solve first-order linear differential equations of the form $$\frac{dy}{dx} + P(x)y = Q(x)$$. 2. The integrating factor, denoted as $$\mu(x)$$,

1. **State the problem:** Solve the differential equation $$2x^3 y' = y(y^2 + 3x^2)$$ for the function $y(x)$. 2. **Rewrite the equation:** Express $y'$ explicitly:

1. **Find the value of $k$ for which $y = e^{kx}$ is a solution to the differential equation** Given differential equation:

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} + \frac{y}{x} = x^2$$ for $y$ as a function of $x$. 2. **Identify the type of equation:** This is a first-

1. **State the problem:** We need to find the general solution of the differential equation $$\frac{dy}{dx} = \frac{y}{x}$$. 2. **Rewrite the equation:** The equation can be writte

1. **State the problem:** Find the differential equation of the family of curves given by $$y = Ae^{2x} + Be^{-2x}$$ where $A$ and $B$ are arbitrary constants. 2. **Differentiate t

1. **State the problem:** Solve the differential equation $$dx + e^{3x} dy = 0$$ for $y$ as a function of $x$. 2. **Rewrite the equation:** We can write it as $$dx = -e^{3x} dy$$ o