📘 differential equations

1. **Problem Statement:** Find the general solution to the differential equation $$y'' - 2y' + y = \frac{e^t}{t^2 + 1}$$ using the method of variation of parameters. 2. **Step 1: S

1. **Problem 14:** Given the differential equation $ (2xy + y^2) dx + (2x^2 + 3xy) dy = 0 $ and an integrating factor of the form $ \mu(x,y) = x^m y^n $, find $m$ and $n$ so the eq

1. **Problem 11:** Given the differential equation \((2xy + y^2) dx + (2x^2 + 3xy) dy = 0\) and an integrating factor \(\mu(x,y) = x^m y^n\), find \(m\) and \(n\) so the equation b

1. Problem 7: Solve the differential equation $$2x(y + 1)dx - ydy = 0$$ with initial condition $$y = -2$$ when $$x = 0$$. 2. Rearrange the equation: $$2x(y + 1)dx = ydy$$.

1. Problem 7: Solve the differential equation $$2x(y + 1) \, dx - y \, dy = 0$$ with initial condition $$y = -2$$ when $$x = 0$$. 2. Step 1: Rewrite the equation in differential fo

1. **Problem statement:** Solve the differential equation $(a - x) dy - (a + y) dx = 0$. 2. **Rewrite the equation:** We can write it as $(a - x) dy = (a + y) dx$ or equivalently

1. **State the problem:** A radioactive substance decays at a rate proportional to its mass. When the mass is 26 mg, the decay rate is 10 mg per week. We want to find a formula for

1. **State the problem:** Solve the differential equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$$ given the initial conditions $$y(2) = 3$$ and $$\frac{dy}{dx}(2) = 2$$. 2. **Ide

1. **State the problem:** Solve the differential equation $$\cos x \frac{dy}{dx} + 2y \sin x = \cos x + \cos^2 x.$$\n\n2. **Rewrite the equation:** Divide both sides by $\cos x$ (a

1. **State the problem:** Find the general solution to the differential equation $$\left(D^4 + 10D^3 + 66D^2 + 184D + 272\right)y = 0,$$ where $D = \frac{d}{dx}$. 2. **Characterist

1. **Staðhæfing verkefnis:** Við eigum að leysa afleiðujöfnuna $$R \frac{dy}{dt} + \frac{1}{C} y = V_0 e^{-rt}$$ með upphafsskilyrðinu $$y(0) = 0$$. 2. **Formúla og aðferð:** Þetta

1. The problem: Understand the method of reduction of order in differential equations. 2. Reduction of order is a technique used to find a second, linearly independent solution to

1. Staðfesta vandamálið: Við eigum að leysa fyrstu stigs afleiðujöfnur. 2. Fyrsta jöfnan: $y' + y = e^x$

1. Problem 25: Find the differential equation for the curve defined by $$\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$$. Step 1: Differentiate both sides with respect to $x$ implicit

1. Problem 15: Given the general solution $y = mx + 3$, where $m$ is an arbitrary constant, find the differential equation by eliminating $m$. 2. Use the formula for the derivative

1. Problem 10: Given $u = e^x \sin 2y$, find partial derivatives and verify the PDE $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 11$. 2. Problem 11: Giv

1. **Problem:** Show that each given function is a solution of the corresponding differential equation. 2. **General approach:** To verify a solution, substitute the function and i

1. Problem: Verify that each given function satisfies its corresponding differential equation. 2. For each function, we will:

1. Problem: Show that each given function is a solution to the corresponding differential equation. 2. For each function, we will:

1. **State the problem:** Solve the differential equation $$xy^2(p^2 + 2) = 2py^3 + y^3$$ where $$p = \frac{dy}{dx}$$. 2. **Rewrite the equation:** Given $$p = \frac{dy}{dx}$$, the

1. **State the problem:** We need to find the general solution of the differential equation $$y^2 \frac{dy}{dx} = x$$ where $c$ is the constant of integration. 2. **Rewrite the equ