📘 differential equations

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

1. **State the problem:** Solve the differential equation $$xy''' + 2y'' = 0$$ for the function $y(x)$. 2. **Rewrite the equation:** The equation is a linear differential equation

1. **Problem statement:** Solve the differential equation $$(1 - x^2) y'' - 2 x y' + 2 y = 0$$ using the Cauchy-Euler (also called Cauchy-Legendre) method. 2. **Recall the Cauchy-E

1. **Problem Statement:** Given the differential equation $$\frac{dy}{dt} = ky$$ where $k$ is a constant and $t$ is in years, and the bacteria doubles every 5 days, find the value

1. **State the problem:** Calculate the Wronskian of the functions $y_1 = 5^x$ and $y_2 = 4x^2$, and determine if they are linearly independent or dependent. 2. **Recall the Wronsk

1. **Problem statement:** We have the third-order homogeneous differential equation $$\frac{d^3x}{dt^3} - 9 \frac{d^2x}{dt^2} + 26 \frac{dx}{dt} - 24x = 0$$ with initial conditions

1. **State the problem:** We want to approximate the value of $y$ at $x=1.1$ given that $y=1.2$ at $x=1$ and the differential equation $$\frac{dy}{dx} = 3x + y^2.$$ We will use Run