📘 differential equations

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

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