📘 differential equations

1. We are given the differential equation: $$\cos(x)\cos(y)\,dx + \sin(x)\sin(y)\,dy = 0$$ 2. To find the general solution, first rewrite in the form: $$M(x,y)\,dx + N(x,y)\,dy = 0

1. **Problem statement:** Solve the differential equation $$\cos(x) \cos(y) \, dx + \sin(x) \sin(y) \, dy = 0.$$\n\n2. **Rewrite and rearrange:** We want to express \( \frac{dy}{dx

1. **Problem Statement:** Find the general solution of the differential equation $$\cos(x) \cos(y) \, dx + \sin(x) \sin(y) \, dy = 0$$. 2. **Rewrite the equation:** We have

1. Stating the problem: Solve the differential equation \(\cos(x) \cos(y) \, dx + \sin(x) \sin(y) \, dy = 0\). 2. Rewrite the equation as:

1. **Problem Q1A:** Solve $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^x \ln x$$ by variation of parameters. - The complementary equation is $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = 0$

1. Solve differential equations by the method of variation of parameters. A. Given: $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^x \sin x$$

1. **State the problem:** We need to find the value of $k$ such that $y = e^{kx}$ is a solution to the differential equation $$y'' - 7 y' + 14 y - 8 y = 0.$$\n\n2. **Simplify the d

1. **State the problem:** Solve the differential equation $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + y = \log x \cdot \sin(\log x).$$ 2. **Rewrite using substitution:** Let $$t =

1. Stating the problem: Solve for $y$ in the equation $$(D^2 + 9)y = (x^2 + 1)e^{3x} \sin x.$$ Here, $D$ represents differentiation with respect to $x$, so $D^2 y = \frac{d^2y}{dx^

1. **State the problem:** We need to solve the differential equation $$G'[\Xi] = \Delta (\Lambda G[\Xi]^3 - G[\Xi]) + \mu.$$\n\n2. **Rewrite the equation:** The equation can be wri

1. **Problem (A):** Given $x=c_1\cos t + c_2\sin t$ solves $x''+x=0$, find $c_1, c_2$ satisfying initial conditions $x(\pi/2)=0$ and $x'(\pi/2)=1$. 2. Differentiate $x(t)$ to get $

1. **State the problem:** We need to find the integrating factor of the differential equation $$(x + 2y^3) \frac{dy}{dx} = 2y.$$ 2. **Rewrite the equation:** Divide both sides by $

Solve differential equations 2 to 7 step-by-step. 2. Given $$x \frac{dy}{dx} = y - \sqrt{x^2 + y^2}$$

1. **Problem:** Solve the differential equation $$2xy \, dx + (y^2 - x^2) \ dy = 0.$$ 2. **Rewrite in differential form:**

1. The problem gives the differential equation $$x y' = y + 2x^3 \sin^2\left(\frac{y}{x}\right).$$ 2. We need to find the solution or explore this equation step by step.

1. The problem asks to solve an Initial Value Problem (IVP), but the differential equation and initial condition are not specified. 2. To solve an IVP, typically you need a differe

1. Given the differential equation: $$xy' = y^2 + y$$ 2. Rewrite it as: $$xy' = y(y+1)$$ which implies $$y' = \frac{y(y+1)}{x}$$

1. The problem is to solve the differential equation $$xy' = y^2 + y$$ or equivalently $$x \frac{dy}{dx} = y^2 + y$$. 2. Divide both sides by $x$ (assuming $x \neq 0$) to rewrite t

1. **State the problem:** We are given the differential equation $$y'' + \lambda y = 0$$ with boundary conditions $$y'(0) = 0$$ and $$y(1) = 0$$. We need to find the eigenvalues \(

1. The problem is to solve the differential equation $$y'' + \lambda y = 0$$ with boundary conditions $$y'(0) = 0$$ and $$y(1) = 0$$. 2. Start by considering the characteristic equ

1. **State the problem:** Solve the differential equation $$y'' + \lambda y = 0$$ with boundary conditions $$y'(0) = 0$$ and $$y(1) = 0$$. 2. **Write characteristic equation:** The