📘 differential equations

# Differential Equations Lecture ## PART 1: BASIC CONCEPTS OF DIFFERENTIAL EQUATIONS

1. **Problem Statement:** Find the charge $q(t)$ and current $i(t)$ at any time $t$ for an electric circuit with inductance $L=0.1$ H, resistance $R=20$ ohms, and capacitance $C=25

1. **Problem 1: Use Euler's method to approximate $y(0.2)$ and $y(0.4)$ for the initial value problem** Given:

1. **State the problem:** We want to find the rate at which salt leaves the tank at time $t$.

1. **State the problem:** We have a tank with an initial volume of 100 gallons containing 70 pounds of salt. Brine with 2 pounds of salt per gallon flows in at 6 gallons per minute

1. **State the problem:** Solve the differential equation $$y'' - 2y - 3y = e^{2t} + 3t^2 + 4t - 5 + 5 \cos(2t)$$ using the Principle of Superposition. 2. **Rewrite the equation:**

1. **State the problem:** Solve the first-order linear differential equation $$y' - 3x^2 y = \frac{6 e^{x^3} (\ln x)^2}{x}, \quad x > 0.$$\n\n2. **Identify the type and formula:**

1. **Problem statement:** Given the second order linear homogeneous differential equation $$y'' + p(x)y' + q(x)y = 0$$ with continuous functions $p$ and $q$ on an open interval $I$

1. **Problem Statement:** Solve a homogeneous differential equation. 2. **Definition:** A first-order differential equation is homogeneous if it can be written as $$\frac{dy}{dx} =

1. **State the problem:** We are given the differential equation:

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{-8e^x \cos(y) + \frac{y^2}{x}}{-8e^x \sin(y) - 2y \ln(x) + 3y^2}$$

1. **State the problem:** We need to solve the first-order linear differential equation:

1. **State the problem:** We have the initial value problem (IVP):

1. **State the problem:** We need to solve the differential equation $$\frac{d y}{d x} = 3 x^2 y$$. 2. **Identify the type of differential equation:** This is a first-order separab

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = -3y - \frac{2}{8x+5}$$. 2. **Rewrite the equation:** Write it in standard linear form:

1. **State the problem:** We need to solve the initial value problem given by the differential equation $$\frac{dy}{dx} = x^2 (y - 2)$$ with the initial condition $$y(0) = 5$$.

1. **State the problem:** We need to solve the initial value problem given by the differential equation

1. **State the problem:** We want to prove that the set of functions {$e^{ax}$, $e^{bx}$} is linearly independent on $\mathbb{R}$ using the Wronskian method. 2. **Recall the Wronsk

1. **Planteamiento del problema:** Resolver la ecuación diferencial de Cauchy-Euler $$x^2y'' + 3xy' + 5y = 0$$ usando el método de Cauchy-Euler. 2. **Fórmula general y método:** Pa

1. **Problem (a): Solve the shifted initial value problem using Laplace transform** Given:

1. **State the problem:** Solve the second-order linear nonhomogeneous differential equation $$y'' + 3y' - 4y = 6e^{2x} - 3$$ with initial conditions $$y(1.5) = 4$$ and $$y'(1.5) =