📘 differential equations

1. **State the problem:** We need to find the general solution of the differential equation $$\frac{dy}{dx} = e^{2x - 5y}$$. 2. **Rewrite the equation:** The equation can be writte

1. The problem is to understand the concept of an exact differential equation and how to solve it. 2. An equation of the form $$M(x,y) + N(x,y)\frac{dy}{dx} = 0$$ is exact if there

1. **State the problem:** We are given the differential equation $$\cos(y) \frac{dy}{dx} = x^2 - 2$$ and asked to:

1. The problem asks which differential equations can be solved using the method of separation of variables. 2. Separation of variables works when the equation can be written as $$\

1. **State the problem:** Solve the differential equation $$(3x^3 - 3x + 1) \frac{dy}{dx} = (3x^2 - 1) y.$$\n\n2. **Rewrite the equation:** We want to express it in the form $$\fra

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = \tan^2(x)(3y - 1)e^y$$. 2. **Identify the type of equation:** This is a separable differential equation

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = \frac{\sin(2x)}{\cos(y) \sin^{-1}(y)}.$$\n\n2. **Rewrite the equation:** We can write the equation as $$

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = e^{-2y} \ln(2x)$$. 2. **Identify the type of equation:** This is a separable differential equation becau

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = y^3 \sec(3x)$$ and express $y$ explicitly. 2. **Identify the type:** This is a separable differential eq

1. Problem: Solve the differential equation $(xy + x) dx = (x^2 y^2 + x^2 + y^2 + 1) dy$. Step 1: Rewrite as $M dx + N dy = 0$ where $M = xy + x$ and $N = -(x^2 y^2 + x^2 + y^2 + 1

1. Solve the differential equation $(xy + x) \, dx = (x^2 y^2 + x^2 + y^2 + 1) \, dy$. Rewrite as $(x(y+1)) \, dx = (x^2 y^2 + x^2 + y^2 + 1) \, dy$.

1. **State the problem:** Solve the differential equation $ (xy + x)\,dx = (x^2y^2 + x^2 + y^2 + 1)\,dy $. 2. **Rewrite the equation:** Divide both sides by $dy$ (assuming $dy \neq

1. The differential equation is $$9y^{(4)} - 12y'' + 4y = 0$$ with initial conditions $$y(0) = 1, y'(0) = 2, y''(0) = 3, y'''(0) = 4.$$\n2. Assume a solution of the form $$y = e^{r

1. **State the problem:** Solve the differential equation $$9y^{(4)} - 12y'' + 4y = 0$$ with initial conditions $$y(0) = 1, y'(0) = 2, y''(0) = 3, y'''(0) = 4.$$\n\n2. **Form the c

1. The problem is to find the general solution of a differential equation or an equation type, but since the specific equation is not provided, we will explain the general approach

1. **State the problem:** Solve the differential equation $$9y^{(4)} - 12y'' + 4y = 0$$ with initial conditions $$y(0) = 1, y'(0) = 2, y''(0) = 3, y'''(0) = 4.$$\n\n2. **Form the c

1. **State the problem:** Solve the differential equation $$y'' + 6y' + 9y = 0$$ with initial conditions $$y(0) = 2$$ and $$y'(0) = -10$$. 2. **Find the characteristic equation:**

1. Problem a) Solve the Bernoulli differential equation $$\frac{dy}{dx} - x y = e^{-x^2} y^3$$ Step 1: Identify the Bernoulli form: $$\frac{dy}{dx} + P(x) y = Q(x) y^n$$ with $$P(x

1. **State the problem:** Solve the differential equation $$y(1 - xy^2)\,dx + dy = 0.$$\n\n2. **Rewrite the equation:** Express it in the form $$M(x,y)\,dx + N(x,y)\,dy = 0$$ where

1. **State the problem:** Solve the differential equation $ (2y + \sqrt{y} \sin x) \, dx + x \, dy = 0 $. 2. **Rewrite the equation:** The given equation is $ (2y + \sqrt{y} \sin x

1. **State the problem:** Solve the differential equation $$(\tan x - \sin x \sin y)\,dx + (\cos x \cos y)\,dy = 0.$$\n\n2. **Rewrite the equation:** We have $M(x,y) = \tan x - \si