📘 differential equations

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 $$\frac{dy}{dx} = 3x + y^2.$$\n\n2. **Recall

1. **State the problem:** We want to solve the differential equation $$y' = 1 - y$$ with initial condition $$y(0) = 0$$ using the modified Euler's method (also known as Heun's meth

1. **Problem statement:** Solve the initial value problem $y' = 1 - y$, with $y(0) = 0$, using the modified Euler's method (also called Heun's method) to find $y$ at $x = 0.1, 0.2,

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = \frac{2x + 4y + 3}{2y + x + 1}.$$\n\n2. **Analyze the equation:** This is a first-order diffe

1. **State the problem:** We want to approximate the value of $y$ at $x=0.1$ for the differential equation $$\frac{dy}{dx} = x + y + xy$$ with initial condition $y(0) = 1$ using Eu

1. **State the problem:** We want to approximate the solution to the differential equation $$y' = x + y$$ with initial condition $$y(0) = 0$$ using Euler's method with step size $$

1. **Problem:** Find the general solution of the differential equation $$(x^2 + 1)dx + xye^y dy = 0.$$ 2. **Step 1: Identify the type of differential equation.**

1. **Problem Statement:** Prove the differential equations:

1. **Stating the problem:** A metal bar initially at temperature $100$°F is placed in a room at constant temperature $30$°F. After $20$ minutes, the bar's temperature drops to $60$

1. **Stating the problem:** We are given the first-order differential equation:

1. **State the problem:** Solve the differential equation $$y(4) - 2y(3) + y'' = e^x + x + \sin x$$ where $y(4)$ is the fourth derivative of $y$, $y(3)$ the third derivative, and $

1. **Problem Statement:** Determine whether the differential equation $\left(3x^2 y - y\right) dx + \left(x^3 - x\right) dy = 0$ is exact or not by evaluating the partial derivativ

1. **Problem Statement:** Determine whether the differential equation $ (3x^2 y - y) \, dx + (x^3 - x) \, dy = 0 $ is open or closed by evaluating the partial derivatives and class

1. **State the problem:** Solve the differential equation $$x (2yx^2 - 3) dy + (3y^2 x^2 - 3y + 4x) dx = 0.$$ 2. **Identify functions:** Let $$M = x(2yx^2 - 3) = 2yx^3 - 3x$$ and $

1. **Problem statement:** Given one solution of a second-order linear differential equation, find the second solution using the reduction of order formula. 2. **Formula:** If $y_1$

1. **Problem statement:** Solve the differential equation $y' - \frac{1}{x} y = 3x$ with the initial condition $y(1) = 5$. 2. **Formula and method:** This is a first-order linear d

1. **Problem Statement:** Find the order and degree of the differential equation $$\frac{d^3 y}{dx^3} + \left(\frac{d^2 y}{dx^2}\right)^{10} + 3 \left(\frac{dy}{dx}\right)^7 + 8y =

1. បញ្ហា៖ ដោះស្រាយសមីការឌីផេរ៉ង់ស្យាល់ $y'' - 2y' - 3y = 0$ ជាមួយលក្ខ័ណគោលដេម $y(0) = 3$ និង $y'(0) = 1$។ 2. សមីការនេះជាសមីការឌីផេរ៉ង់ស្យាល់លំដាប់ទីពីរ ដែលមានរាង $ay'' + by' + cy =

1. Problem: Solve the differential equation $$y'' - 9y = 0$$ with initial conditions $$y(\ln 2) = 1$$ and $$y'(\ln 2) = 3$$. 2. The characteristic equation is $$r^2 - 9 = 0$$.

1. **State the problem:** Solve the ordinary differential equation $$y'' + 16y = 0$$ where $y''$ denotes the second derivative of $y$ with respect to $x$. 2. **Identify the type of

1. **Problem:** Determine if the differential equation $$3x^2y - y \, dx + x^3 - x \, dy = 0$$ is open or closed by evaluating partial derivatives, and classify it as exact, separa