📘 differential equations

1. **State the problem:** Solve the differential equation $$y'' + 4y = 4 \sin(2t)$$ and find a particular solution $$y_p$$. 2. **Identify the type of equation:** This is a second-o

1. **Problem statement:** Find a particular solution $y_p$ to the nonhomogeneous differential equation $$y'' + 4y' + 5y = -15x + e^{-x}.$$ Then find the general solution $y_h$ to t

1. **State the problem:** We need to find the particular solution for the differential equation $$\frac{dy}{dx} = \frac{3x + y}{x}$$. 2. **Rewrite the equation:** Simplify the righ

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \sqrt{x y^2 + 2 y} = 3 x^2 y^7 + \sin(x y)$$ and need to analyze or solve it. 2. **Understand the

1. **State the problem:** Solve the differential equation $$x\,dy + (y - y^2 x \ln x)\,dx = 0.$$\n\n2. **Rewrite the equation:** We have $$x\,dy + (y - y^2 x \ln x)\,dx = 0,$$ whic

1. Problem: Solve the differential equation $$\frac{dy}{dx} = e^{x - y}$$ Step 1: Rewrite the equation as $$\frac{dy}{dx} = e^x e^{-y}$$.

1. **Problem Statement:** (a) Explain the differences between:

1. **State the problem:** We are given the system of differential equations:

1. Solve $y'' + 4y = e^{3x}$ using undetermined coefficients. - Characteristic equation: $r^2 + 4 = 0 \Rightarrow r = \pm 2i$.

1. **State the problem:** Given the differential equation $x \frac{dy}{dx} - y = 3$ with the initial condition $x=1$ when $y=-2$, find the relation between $x$ and $y$. 2. **Rewrit

1. **State the problem:** We are given the differential equation $$y \sqrt{x + 1} y' = y + 3$$ and need to solve for $y$ as a function of $x$.

1. **Problem Statement:** Find the correct particular integral (PI) for the differential equation $$ (D^3 - D^2 - 6D)y = x^2 + 1 $$ where $D$ represents differentiation with respec

1. **Problem Statement:** Solve the differential equation $$\left(D^6 - 64\right)y = 0$$ where $D$ denotes differentiation with respect to $x$. 2. **Characteristic Equation:** Repl

1. **Stating the problem:** We are given the differential equation $$\frac{dr}{d\theta} = 500\theta^n - \frac{r}{\theta}$$ and asked to find the integrating factor.

1. **State the problem:** We are given the differential equation $$y(xy + e^x)dx - e^x dy = 0$$ and asked to determine which of the given functions is an integrating factor. 2. **R

1. **Stating the problem:** We are given a differential equation of the form $$P(y) \, dx + Q(x) \, dy = 0$$ where $P(y)$ is a function of $y$ only and $Q(x)$ is a function of $x$

1. **State the problem:** We are given the differential equation $$2y \frac{dy}{dx} + \tan(xy) = \frac{(4x + 5)^2}{\cos x} y^3.$$ We want to analyze or solve this equation.

1. **State the problem:** Given the function $y = ax + b/x$, show that it satisfies the differential equation $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - y = 0.$$\n\n2. **Recall t

1. **Stating the problem:** We are given the boundary value problem $$-y^{(4)} + p y = x^4 (32^2 x (-6(7 - 55 x^4 + 70 x^8) + 2 (x^2 - 3 x^6 + 2 x^{10})) \cos x + x^4 (x^4 - 1)^2 -

1. Identify the order and degree of the differential equations: 1.a. Given equation: $ (y'')^{-2} + y' = xy'' + \sin x $

1. Solve the differential equation $y'' + 14y' + 49y = 0$ with initial conditions $y(-4) = -1$ and $y'(-4) = 5$. - This is a second-order linear homogeneous differential equation w