📘 differential equations

1. **State the problem:** Solve the differential equation $$2xyy' = 3y^2 + x^2$$ with the initial condition $$y(1) = 2$$. 2. **Rewrite the equation:** We have $$2xy \frac{dy}{dx} =

1. **Problem Statement:** Solve the differential equation using the method of separable variables. 2. **General Approach:** A separable differential equation can be written as $$\f

1. **State the problem:** We need to find the general solution of the differential equation $$ yy'' = (y')^2 (1 - y' \sin y - y y' \cos y) $$

1. **State the problem:** Find the general solution to the differential equation $$(\sin y - 2y e^x \sin x)\,dx + (\cos y + 2 e^x \cos x)\,dy = 0.$$\n\n2. **Check if the equation i

1. **State the problem:** We are given the differential equation:

1. **Problem Statement:** We are given the differential equation $$2y'' + 18\pi^2 y' = f(t)$$ where $$f(t)$$ is a periodic odd function defined as: $$f(t) = \begin{cases} -1 & -1 \

1. **Problem Statement:** Solve the differential equation $$ (3x^2 y^3 + 5xy) \, dx + (x^4 y^3 - x^4) \, dy = 0.$$\n\n2. **Check if the equation is exact:**\nLet \(M = 3x^2 y^3 + 5

1. Das Problem besteht darin, die allgemeine reelle Lösung des Differenzialgleichungssystems $$\dot{x}(t) = \begin{pmatrix} 2 & 52 \\ -1 & -10 \end{pmatrix} x(t)$$ zu bestimmen, wo

1. **Énoncé du problème** : Résoudre l'équation différentielle $y' - 2y = \cos x + 2 \sin x$. 2. **Formule et méthode** : C'est une équation différentielle linéaire du premier ordr

1. The problem is to find the general solution of the equation $$p=\cos(y-px)$$ where $p=\frac{dy}{dx}$. 2. This is a first-order differential equation involving $p$ and $y$. We wa

1. The problem is to determine the condition under which the differential equation $M(x,y)dx + N(x,y)dy = 0$ is exact. 2. An exact differential equation means there exists a functi

1. The problem is to find the integral of the function $$\frac{\cos x}{D^2+1}$$ where $D$ is the differential operator $\frac{d}{dx}$. This is interpreted as solving the differenti

1. **Stating the problem:** Simplify or understand the expression $$\frac{1}{D^2+1} \cos x$$ where $D$ represents the differentiation operator with respect to $x$. 2. **Understandi

1. **State the problem:** We are given the second-order differential equation $$\frac{d^2y}{dx^2} = 2y^3 - y$$ and want to analyze it using substitution and integration techniques.

1. **State the problem:** We need to find the general solution of the ODE given by $$y + [y(x^2 + y^2) - x] \bar{y} = 0$$

1. **Stating the problem:** We need to find $\frac{dy}{dx}$ given the differential equation $$\frac{dy}{dx} - y = e^x y^2.$$ 2. **Rewrite the equation:** Add $y$ to both sides to i

1. Vamos resolver as aproximações para as soluções dos problemas de valor inicial (PVI) usando o método de Euler com passo $h=0,1$. O método de Euler é dado pela fórmula: $$y_{n+1}

1. **Problem Statement:** Find the general solution to the differential equation using variation of parameters, given the nonhomogeneous term $y = t^2$ and the fundamental solution

1. **Problem Statement:** Find the general solution to the differential equation $$y'' + y = \tan(t)$$ using the method of variation of parameters. 2. **Step 1: Solve the homogeneo

1. **Problem statement:** Solve the initial value problem using variation of parameters: $$y'' - 4y = (12x^2 - 6x)e^{2x}, \quad y(0) = 1, \quad y'(0) = 0$$

1. **State the problem:** Find the general solution to the differential equation $$y''' + 4y' = \sec(2x)$$ using the method of variation of parameters. 2. **Find the complementary