📘 differential equations

1. مسئله: حل معادله دیفرانسیل $$x^r y'' - rxy' + ry = x^{-r}$$ است. 2. فرمول‌ها و قواعد: این معادله از نوع معادلات دیفرانسیل مرتبه دوم با ضرایب متغیر است. ابتدا باید معادله همگن مت

1. **Problem Statement:** Solve the initial value problem (IVP) given by the differential equation

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = -y - \sin x$$. 2. **Identify the type of equation:** This is a first-order linear ordinary differential

1. مسئله: یافتن جواب خصوصی معادله دیفرانسیل $$D(D^2+9)(D^2-8D+25)^2 y = x + x^2 \sin^3 x + x^2 e^{-8x} \cos 3x$$

1. **Stating the problem:** Solve the differential equation $$2(y')^2 = (y-1)y''$$ where $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2y}{dx^2}$. 2. **Rewrite the equation:** Let $p = y

1. **Stating the problem:** We are given the differential equation $$y' = e^{x+y} - 1$$ and asked to analyze or solve it.

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = -y - \sin x$$. 2. **Identify the type of equation:** This is a first-order linear ordinary di

1. **State the problem:** Solve the differential equation $$\sin y\,dy + \cos y(1 - \cos y \cos x)\,dx = 0.$$\n\n2. **Rewrite the equation:** The equation is $$\sin y\,dy + \cos y(

1. **Stating the problem:** Solve the differential equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = x \sin x.$$\n\n2. **Identify the type of equation:** This is a nonhomogeneous

1. **State the problem:** Solve the differential equation $$x^2 y'' + 4x y' + 2x = 4 \ln x$$ using the same method (likely variation of parameters or reduction of order). 2. **Rewr

1. **State the problem:** Solve the differential equation $$x^2 y'' - 4x y' + 6y = 4x - 6$$ using the method of variation of parameters or a similar method. 2. **Identify the type

1. **State the problem:** Solve the differential equation $$4x^2y'' - 4xy' + 3y = 0$$ using the Cauchy-Euler method. 2. **Recall the Cauchy-Euler form:** Equations of the form $$x^

1. **State the problem:** Solve the differential equation $$x^2 y'' - 3x y' + 3y = 0$$ using the Cauchy-Euler method. 2. **Recall the Cauchy-Euler form:** The equation is of the fo

1. **State the problem:** Solve the second-order linear differential equation $$y'' - 2y' - 8y = 0$$ with initial conditions $$y(0) = 3$$ and $$y'(0) = 6$$. 2. **Characteristic equ

1. **State the problem:** Solve the differential equation $$y'' - 6y' + 9y = 0$$ using the Laplace transform. 2. **Recall the Laplace transform properties:**

1. **Problem Statement:** Find the inverse Laplace transform $\mathcal{L}^{-1}\{F(s)\}$ for the given functions using the Laplace transform table. ---

1. The problem is to find the inverse Laplace transform of the function $$F(s) = \frac{10s + 23}{5s^2 + 7s + 12}$$ using the Laplace transform table. 2. Recall that the Laplace tra

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} + 3y = 3x^2 e^{-3x}$$ using Bernoulli's method. 2. **Identify the type:** This is a linear first-order dif

1. **Problem statement:** Solve the ordinary differential equation (ODE) $$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = x \sin x.$$\n\n2. **Identify the type of equation:** This is a n

1. **Problem Statement:** Given the function:

1. The problem involves analyzing the function $$\mathbf{x}(t) = e^{-0.266t} \left\{ C_1 \begin{bmatrix} 0.732 \\ 1 \end{bmatrix} \cos 7.96t + C_2 \begin{bmatrix} 0.732 \\ 1 \end{b