📘 differential equations

1. مسئله: حل معادله دیفرانسیل برنولی $y' + \frac{y}{x} = x^4 y^3$. 2. معادله برنولی به شکل کلی است: $y' + P(x)y = Q(x)y^n$ که در اینجا $P(x) = \frac{1}{x}$، $Q(x) = x^4$ و $n=3$.

1. مسئله: حل معادله دیفرانسیل برنولی به شکل $$y' + \frac{y}{x} = x^r y^r$$ است. 2. فرمول و روش حل: معادله برنولی به صورت کلی $$y' + P(x)y = Q(x)y^n$$ است. برای حل، ابتدا متغیر جدید

1. **State the problem:** Solve the differential equation $$y'' - 2y' + y = (x+5)^2$$. 2. **Identify the type of equation:** This is a non-homogeneous linear second-order different

1. **Problem Statement:** Solve the differential equation $$\frac{dy}{dt} = -\frac{y}{t} + 2$$ for the general solution. 2. **Identify the type of equation:** This is a first-order

1. State the problem. We are asked to find the Laplace transform of $3e^{-2t}$.

1. **Problem Statement:** Solve the differential equation $ (2x - y) dx - (x + 4y) dy = 0 $ and find the implicit solution. 2. **Identify the type:** This is a first order differen

1. **State the problem:** Calculate the inverse Laplace transform $x(t)$ given by $$x(t) = \mathcal{L}^{-1} \left\{ \frac{5}{2(s+1)} \right\} - \mathcal{L}^{-1} \left\{ \frac{6}{5(

1. The problem involves solving a system of differential equations with given parameters $\kappa = 0.27$, $b = 10$, $\sigma = \frac{5}{2}$, and $h = 1.2$. The function $g[t]$ is de

1. **Problem:** Write the given differential equations in differential form. 2. **Recall:** The differential form of an equation involving derivatives $y'$ or $\frac{dy}{dx}$ is ty

1. **State the problem:** Solve the differential equation $ (1 + y \tan(x)) \, dy - (1 + \cos(x)) \, dx = 0 $. 2. **Rewrite the equation:** Rearrange terms to isolate $dy$ and $dx$

1. **Problem:** Solve the differential equation $$(D^3 + 3D^2 + 3D + 1) y = 0$$ where $D = \frac{d}{dx}$. 2. **Characteristic equation:** Replace $D$ by $r$ to get

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

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = \frac{xy - 3x - y - 3}{xy - 2x - 4y - 8}$$ using separable variables. 2. **Rewrite the equation:** Facto

1. **Problem statement:** Solve the initial value problem (IVP) given by the differential equation $$y' = 10 \sin(x^2 + y^2)$$ with initial condition $$x(0) = 1$$ and step size $$h

1. **State the problem:** Find the general solution of the differential equation $$y''' - 4y'' - 5y' = 0$$. 2. **Identify the type of equation:** This is a linear homogeneous diffe

1. **State the problem:** Solve the differential equation $$y'' - 4y' - 5y = 0$$ using the method of undetermined coefficients. 2. **Identify the type of equation:** This is a line

1. **State the problem:** Find the general solution of the differential equation $$x^2 \frac{dy}{dx} = y - xy$$. 2. **Rewrite the equation:** Divide both sides by $x^2$ (assuming $

1. Problem 5: Find the integrating factor $\mu(x,y)$ for the differential equation $$y\,dx + (3 + 3x - y)\,dy = 0$$ to make it exact. 2. Recall that a differential equation $M(x,y)

1. The problem is to find the general solution of the differential equation $$y' + y = x$$. 2. This is a first-order linear differential equation of the form $$y' + p(x)y = q(x)$$

1. **Problem Statement:** Obtain the differential equation from the relation $y = cx + c^2 + 1$. Show that $y_1 = e^x$ and $y_2 = xe^x$ form a fundamental set of solutions of the d

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = \frac{\tan y - 2xy - y}{x^2 - x \tan^2 y + \sec^2 y}.$$\n\n2. **Analyze the equation:** This is a first-