📘 differential equations

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

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 $y_1$