📘 differential equations

1. **Problem 1 (a): Verify Sturm-Liouville system** The differential equation is $$y'' + \lambda y = 0,$$ with boundary conditions $$y(0) = 0,\quad y(1) = 0.$$

1. The order of a differential equation is the highest order derivative present, and the degree is the power of the highest order derivative when the equation is polynomial in deri

1. Determine the order and degree of the D.E.: Given a differential equation, the order is the highest derivative's order and degree is the power of the highest order derivative af

1. Determine the order and degree of a differential equation (D.E.): The order is the highest derivative present in the equation. The degree is the power of the highest order deriv

1. The order of a differential equation (D.E.) is the highest derivative present. The degree is the power of the highest order derivative if the equation is polynomial in derivativ

1. **Problem:** Solve the differential equation $(D^2 + D)y = -\cos x$ using the method of undetermined coefficients. Step 1: Identify the characteristic equation for the complemen

1. **Stating the problem:** We need to solve the differential equation $$y y'' = 2(y')^2 - 2 y'$$ where $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2y}{dx^2}$. 2. **Rewrite the equatio

1. **Stating the problem:** We are given the differential equation $y y'' = 2 y'' - 2 y'$. Our goal is to solve for $y$. 2. **Rearrange the equation:** The given equation is $y y''

1. **State the problem:** Given the differential equation $y y'' = y'' - y'$, we want to simplify and solve this or analyze it. 2. **Rewrite the equation:**

1. **Stating the problem:** Solve the first-order linear differential equation $$y' = 2ty + 3y - 4t - 6$$ for the function $y(t)$. 2. **Rewrite the equation:** Combine like terms o

1. We are given the differential equation $$4t^4y^3 \, dt + (3t^4y^2 + 6y^2) \, dy = 0.$$ 2. Rewrite the equation as $$4t^4 y^3 \, dt + (3t^4 y^2 + 6 y^2) \, dy = 0.$$

1. **Stating the problem**: We want to find methods to solve first-order nonlinear differential equations. 2. **Separating variables**: If the equation can be written as $$\frac{dy

1. Problemi qeyd edək: verilmişdir iki fərqli tənlik. 2. Birinci tənlik: $$61 x \frac{dx}{dt} + t = 1$$.

1. **Problem statement:** Solve the differential equation $$y'' + 4y' + 5y = x + 2$$

1. **Problem:** Solve the ODE $$y'' + 2y' + y = e^x$$ by variation of parameters. 2. **Find the complementary solution:** Solve the homogeneous equation $$y'' + 2y' + y = 0$$.

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** $$ (e^{x+y} + y e^{y}) \, dx + (x e^{y} - 1) \, dy = 0, \quad y(0) = -1 $$