📘 differential equations

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

1. **Problem Statement:** Find the isogonal trajectories that cut at an angle of 45 degrees to the family of lines passing through the origin. 2. **Given Family:** The family of li

1. **Problem 1: Compute the Wronskian for** $$\vec{y}_1(t) = \begin{bmatrix} \cos(2t) \\ -\sin(2t) \end{bmatrix}, \quad \vec{y}_2(t) = \begin{bmatrix} -2 \sin(2t) \\ -2 \cos(2t) \e

1. **State the problem:** Find a particular solution $y_p$ to the differential equation $$y'' + 5y' + 4y = -10te^{3t}.$$\n\n2. **Identify the type of equation:** This is a nonhomog

1. **State the problem:** Find a particular solution $y_p$ to the differential equation $$y'' + 5y' + 4y = -10te^{3t}.$$\n\n2. **Identify the type of equation:** This is a nonhomog

1. **Problem 1:** Solve the differential equation $ (1 - x) dy + (1 - y) dx = 0 $. 2. Rewrite the equation as $$ \frac{dx}{1 - x} + \frac{dy}{1 - y} = 0 $$ to separate variables.

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