📘 differential equations

1. **State the problem:** Solve the differential equation $$3\,dx + e^{3x}\,dy = 0$$. 2. **Rewrite the equation:** We can write it as $$3\,dx = -e^{3x}\,dy$$ or $$\frac{dy}{dx} = -

1. **Form the differential equation from** $y = x + \frac{A}{x}$. Step 1: Differentiate $y$ w.r.t. $x$.

1. **Problem:** Cane sugar in water converts to dextrose at a rate proportional to remaining amount. Of 75 kg, 0.8 kg converts in first 30 minutes. Find amount converted in 2 hours

1. We are given the initial-value problem $$y' = x^2 y - \frac{1}{2} y^2,\quad y(0)=9$$ and we need to estimate $$y(1)$$ using Euler's method with step size $$h=0.2$$. 2. Recall Eu

1. Problem 1(a): Solve the differential equation $$\sin(2x) \frac{dy}{dx} = y \sin x$$ with initial condition $$y(0) = 2$$. Step 1: Rewrite the equation as $$\frac{dy}{dx} = \frac{

1. **State the problem:** Solve the differential equation $$\cos(x)\cos(y) \, dx + \sin(x)\sin(y) \, dy = 0$$. 2. **Separate variables:** Rewrite terms to isolate $dx$ and $dy$:

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.