📘 differential equations

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

1. **State the problem:** We want to find the most suitable integrating factor by inspection for the differential equation $$y(2x + y)dx - x^2 dy = 0$$ to rewrite it as a derivativ

1. **State the problem:** We are given the differential equation $$y(2x + y)dx - x^2 dy = 0$$ and asked to find the most suitable integrating factor by inspection. 2. **Rewrite the

1. **State the problem:** We want to find the most suitable integrating factor by inspection for the differential equation $$y(2x + y)dx - x^2 dy = 0$$ to make it separable or exac

1. مسئله: معادله دیفرانسیل $$T_2''(t) + 4 T_2(t) = \cos(2t)$$ را حل کنید. 2. فرمول و روش کلی: این معادله یک معادله دیفرانسیل خطی غیرهمگن مرتبه دوم با ضرایب ثابت است. ابتدا معادله ه

1. **Stating the problem:** We have an electric circuit with two loops and currents $i_1(t)$ and $i_2(t)$.

1. **Stating the problem:** We are given the differential equation $$y' = \frac{x^2 + xy}{xy}$$ and asked to write it in homogeneous form. 2. **Recall the definition:** A different

1. مسئله: حل معادله دیفرانسیل با روش جداسازی متغیرها. 2. فرض کنیم معادله به صورت $$\frac{dy}{dx} = g(x)h(y)$$ باشد.

1. **Problem Statement:** We are given a differential equation $$\frac{dP}{dt} = f(P)$$ with initial value $$P(t_0) = P_0$$. We want to find for which positive initial values $$P_0

1. **Problem statement:** Find the general solution to the differential equation $$y'' + 16y = 0$$. 2. **Formula and approach:** This is a second-order linear homogeneous different

1. **Problem:** Use Euler's method to solve the differential equation $$\frac{dy}{dx} = 1 - y$$ with initial condition $$y(0) = 0$$, step size $$h = 0.1$$, over the interval $$[0,

1. **State the problem:** Solve the differential equation $$y'' + 2y' + y = 0$$ with initial conditions $$y(0) = 2$$ and $$y'(0) = 10$$. 2. **Identify the type of equation:** This

1. **Problem Statement:** Find the general solution to the differential equation $$y''' - 3y'' + 2y' = \frac{e^x}{1 + e^{-x}}$$ using the method of variation of parameters. 2. **St

1. **State the problem:** Find the general solution of the differential equation $$\frac{dy}{dx} = xy$$ for $$y > 0$$. 2. **Identify the type of differential equation:** This is a

1. **State the problem:** Solve the differential equation $$y^{(5)} - y^{(4)} = 0$$ where $$y^{(5)}$$ is the fifth derivative of $$y$$ and $$y^{(4)}$$ is the fourth derivative. 2.

1. **State the problem:** Solve the differential equation $$y''''' - y'''' = 0$$. 2. **Identify the type of equation:** This is a linear homogeneous differential equation with cons