📘 differential equations

1. The Airy equation is a second-order linear differential equation defined as: $$ y'' - xy = 0 $$

1. The problem asks about what must be imposed to find a particular solution to a differential equation. 2. In differential equations, general solutions often contain arbitrary con

1. The problem asks to determine the order of the given differential equation. 2. The given differential equation is

1. The given equation is $$x^2 y'' + x y' + (x^2 - \nu^2) y = 0$$. 2. This is a standard form of Bessel's equation of the first kind.

1. The problem asks us to verify if the function \(y = e^{-2x} (\cos 3x + \sin 3x)\) is a solution to the differential equation \(\frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + 13y = 0\).

1. **Problem 1: Solve the initial value problem** $$ (e^{x+y} + y e^{y}) \, dx + (x e^{y} - 1) \, dy = 0, \quad y(0) = -1 $$

1. **Problem a:** Solve the differential equation $$y'=1+e^{y-x+5}$$. Step 1: Rewrite the equation as $$\frac{dy}{dx}=1+e^{y-x+5}$$.

I. Variable Separable Differential Equations 1. Solve $y' = x^{-2}$ passing through $(3,2)$.

**Problem Set 2: Differential Equations** ### I. Variable Separable Differential Equations

1. **State the problem:** Solve the differential equation $$(1 - xy + x^2 y^2) \, dx = (x^2 - x^3 y) \, dy.$$\n\n2. **Rewrite the equation:** Express in form $$M(x,y)\,dx + N(x,y)\

1. The problem is to solve the differential equation: $(1 - xy + x^2 y^2) dx = (x^2 - x^3 y) dy$. 2. Rewrite the equation as $ (1 - xy + x^2 y^2) dx - (x^2 - x^3 y) dy = 0 $.

1. **State the problem:** Solve the differential equation $ (2xy + y^2)\,dx - 2x^2\,dy = 0 $. 2. **Rewrite the equation in differential form:** The equation can be expressed as $ M

1. The problem asks for the value of constant $C$ for the differential equation $$Xy^2 dy - (x^3 + y^3) dx = 0$$ when $y=3$ and $x=1$. 2. Identify if the equation is exact or separ

1. **Problem:** Solve the variable separable differential equation $y' = \frac{x^{-2}}{x^{-2}}$ passing through the point $(3, 2)$. 2. **Rewrite the equation:** Since both numerato

1. State the problem: Solve the second-order linear differential equation $$x'' - 2x' + x = 2$$ where $x''$ is the second derivative of $x$ with respect to $t$, and $x'$ is the fir

1. نبدأ ببيان المسألة: نحن نبحث عن الحل الوحيد للمعادلة التفاضلية $$y' = y$$ مع الشرط الابتدائي $$y(0) = 1$$. 2. لحل المعادلة التفاضلية: المعادلة هي من نوع المعادلات التفاضلية الخط

1. Problem 1: Find the integrating factor for the differential equation $$dx + \left(\frac{x}{y} - \sin y\right) dy = 0.$$ - Generally, the integrating factor depends on whether th

1. **Solve the D.E.** \((2x^3 - xy^2 - 2y + 3) dx - (x^2 y + 2x) dy = 0\). - Check if the equation is exact by verifying \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partia

1. Let's examine the first differential equation (D.E.): $$(3x^2y - 6xy) dy + (x^3 + 2y) dx = 0.$$ We want to check if this equation is exact or if an integrating factor is needed.

1. The problem is to understand what a differential equation is and how to solve a simple example. 2. A differential equation is an equation involving derivatives of a function. It

1. Solve $y'' - y' - 6y = 0$. Step 1: Write the characteristic equation: $$r^2 - r - 6 = 0$$.