📘 differential equations

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

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 $$y - x \frac{dy}{dx} = a \left(y^2 + \frac{dy}{dx}\right).$$ 2. **Rewrite the equation:** Group terms involving $\frac{dy

1. **State the problem:** Solve the differential equation $$y - x \left(\frac{dy}{dx}\right) = a \left(y^2 + \frac{dy}{dx}\right).$$

1. **State the problem:** Solve the differential equation $ (x + y e^{\frac{y}{x}}) \, dx - x e^{\frac{y}{x}} \, dy = 0 $ with initial condition $ y(1) = 0 $.\n\n2. **Rewrite the e

1. **State the problem:** Solve the differential equation $$\left(x + ye^{\frac{y}{x}}\right) dx - xe^{\frac{y}{x}} dy = 0$$ with initial condition $$y(1) = 0.$$\n\n2. **Rewrite th

1. **State the problem:** Solve the differential equation $$2x\,dx + \frac{x^2}{y}\,dy = 0$$ by finding an integrating factor and then solving the equation. 2. **Rewrite the equati

1. **State the problem:** We are given the differential equation $$2x\,dx + x^2 y\,dy = 0$$ and asked to find an integrating factor and solve it.

1. **Problem statement:** We consider the odd-order linear differential equation $$y^{(2n+1)} + \sum_{k=0}^{2n} a_k y^{(k)} = 0, \quad x \in (0,L),$$

1. Tentukan solusi trayektori ortogonal dari keluarga kurva: **a.** $5x^2 + y = \lambda$

1. **Problem Statement:** Solve the first order differential equation $\frac{dy}{dx} = y$. 2. **Formula and Explanation:** This is a separable differential equation. The general fo

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = e^{x+y}$$. 2. **Rewrite the equation:** Use the property of exponents to separate variables:

1. **Problem:** Solve the first order linear differential equation $$\frac{dy}{dt} + 4y = 12$$ with initial condition $$y(0) = 2$$. 2. **Formula and rules:** The general form of a

1. **State the problem:** Find the inverse Laplace transform of $$\frac{a \cos(b) - s \sin(b)}{s^2 + a^2}$$. 2. **Recall the Laplace transform formulas:**

1. **Problem 1:** Solve the differential equation $$\frac{d^2x}{dt^2} - 2 \frac{dx}{dt} + 5x = 34 \cos 2t$$ with initial conditions $$x(0) = 10$$ and $$x'(0) = 2$$. 2. **Problem 2:

1. Problem 4a: Find the inverse Laplace transform of $$\frac{4s - 2}{s^2 - 6s + 18}$$. 2. First, complete the square in the denominator:

1. **Problem statement:** Solve the first-order linear differential equation $$y' + 2ty = 2te^{-t^2}$$

1. The problem is to solve the differential equation $$x'' + 4x' + 5x = 80\sin 5t$$ with initial conditions $$x(0) = 0$$ and $$x'(0) = 0$$. 2. This is a second-order linear nonhomo