📘 differential equations

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: $$r^2 - r - 6 = 0$$.