📘 differential equations

1. **State the problem:** Find the differential equations of the space curves formed by the intersection of the two families of surfaces given by: $$u = x^2 + y^2 + z^2 = c_1$$

1. **State the problem:** Solve the initial value problem $$\frac{dy}{dx} = \frac{y^2 - \cos x \sin x}{x(1 - x^2)}$$ with initial condition $$y(0) = 2$$. 2. **Analyze the different

1. Problem: Solve the differential equation $xy \, dx + y^2 \, dy = 0$. 2. Formula and rules: This is a separable differential equation. We can rewrite it to separate variables $x$

1. The problem is to identify whether a given differential equation (DE) is an ordinary differential equation (ODE) or a partial differential equation (PDE), and to state its order

1. **Problem A1:** Solve the differential equation $$y'' - 5y' + 6y = 0.$$ 2. **Formula and rules:** This is a second-order linear homogeneous differential equation with constant c

1. **Problem Statement:** We have water flowing from an inverted conical tank with the rate of change of water height given by

1. **State the problem:** Solve the system of differential equations given by $$\mathbf{x}' = A\mathbf{x}$$ where $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ and $$A

1. The problem is to eliminate the arbitrary constants $A$ and $B$ from the function $$y = Ax^2 + Be^{2x}$$. 2. This function is a linear combination of two independent functions:

1. **State the problem:** We are given the function $$y = Ae^{2x} + Bxe^{2x}$$ where $A$ and $B$ are arbitrary constants.

1. **Problem:** Eliminate the arbitrary constant $C$ from the equation $$y = Cx + C^2 + 1$$ 2. **Step 1: Differentiate both sides with respect to $x$**

1. **Problem 1: Solve the differential equation** Given: $$y' = y - \frac{2x}{y}, \quad y(0) = 1$$

1. **State the problem:** A tank initially contains 100 gallons of fresh water. Saltwater with concentration $\frac{1}{2}$ lb/gal flows in at 2 gal/min, and the mixture flows out a

1. Diberikan persamaan diferensial koefisien linier nonhomogen: $$ (a_1 x + b_1 y + c_1) dx + (a_2 x + b_2 y + c_2) dy = 0 $$

1. **State the problem:** Solve the partial differential equation (PDE) $$x^2 r + 2 x y s + y^2 t = 0$$ where $r = \frac{\partial^2 z}{\partial x^2}$, $s = \frac{\partial^2 z}{\par

1. **Nyatakan masalah:** Selesaikan persamaan pembezaan linear tak seragam $$3y'' + 7y' + 2y = (x+1)e^{-2x}$$

1. **Problem statement:** Solve the second order differential equation $$y'' - 2y' + y = e^x$$. 2. **Find the complementary solution $g(x)$:**

1. **State the problem:** Solve the differential equation $$\sqrt{x+y+1} \frac{dy}{dx} = x + y - 1.$$\n\n2. **Rewrite the equation:** Let $z = x + y$. Then $y = z - x$ and $$\frac{

1. **Problem:** Solve the differential equation $$\frac{dy}{dx} - \frac{y}{x} = xy^2$$ using Bernoulli's equation. 2. **Bernoulli's equation form:** $$\frac{dy}{dx} + P(x)y = Q(x)y

1. **State the problem:** Solve the differential equation $$(x^2-1)y'' + xy' - y = 0.$$\n\n2. **Identify the type:** This is a second-order linear differential equation with variab

1. **State the problem:** Solve the differential equation $$(x - y)(4x + y)\,dx + x(5x - y)\,dy = 0.$$\n\n2. **Rewrite the equation:** The given equation is $$(x - y)(4x + y)\,dx +

1. **State the problem:** Solve the differential equation $$(x - y)(4x + y) \, dx + x(5x - y) \, dy = 0.$$ We want to find the general solution and identify which of the given opti