📘 differential equations

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

1. The problem is to write down the Cauchy-Euler differential equation, commonly studied in civil engineering B.Tech courses. 2. The Cauchy-Euler differential equation is a type of

1. The problem is to write down the Cauchy-Euler differential equation. 2. The Cauchy-Euler differential equation is a type of linear differential equation with variable coefficien

1. **State the problem:** We are given a differential equation and need to determine if it is open or closed by evaluating the partial derivatives. Then, classify the equation as e

1. Let's start by stating the problem: What is an exact differential equation? 2. An exact differential equation is a first-order differential equation that can be written in the f

1. **State the problem:** Solve the differential equation $$y'' - 2y' - 3y = 0$$ with initial conditions $$y(0) = 3$$ and $$y'(0) = 1$$. 2. **Characteristic equation:** For a linea

1. **Problem statement:** We are given two piecewise functions to analyze and understand:

1. **Stating the problem:** We need to find the function $b(r)$ given the equation

1. **Stating the problem:** We need to find the function $b(r)$ satisfying the equation

1. **State the problem:** We are given the equation $$b(r)\frac{\alpha}{r^3} + E^2 = w \left(-\alpha \frac{b'(r)}{r^2} - E^2\right)$$ and need to analyze or solve it.

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

1. **State the problem:** We need to solve the equation $$b(r)\frac{\alpha}{r^3} + E^2 = -w b(r)^u \left(-\alpha \frac{b'(r)}{r^3} - E^2\right)$$ for the function $b(r)$. 2. **Rewr

1. Find the Laplace Transforms of the following functions: (a) $f(t) = 2t^3 + 4 \cos 5t$

1. **State the problem:** Find the general solution of the differential equation $$y' = e^{8x} - 7x$$. 2. **Recall the formula:** The general solution of a first-order differential

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = (x-4)e^{-2y}$$ with the initial condition $$y(4) = \ln(4)$$. We want to find the explicit solutio

1. **State the problem:** We are given the differential equation $$\frac{dP}{dt} = 3P + a$$ where $a$ is a constant. 2. **Identify the type of equation:** This is a first-order lin