📘 differential equations

1. **Problem:** Solve the initial value problem $y' = 10 - x$, with $y(0) = -1$. 2. **Formula and Explanation:** This is a first-order ordinary differential equation (ODE). The gen

1. **State the problem:** Solve the differential equation $$\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0.$$\n\n2. **Rewrite the equation:** Let $$M(x,y) = \sec^2 x \tan y$$ an

1. **Problem 8:** Form the differential equation representing the family $$y = \frac{C_1 e^{3x} + C_2 e^{-3x}}{C_3 \cos x + C_4 \sin x}.$$

1. Let's state the problem: Solve a differential equation using the Laplace transform method. 2. The Laplace transform of a function $f(t)$ is defined as $$\mathcal{L}\{f(t)\} = F(

1. **Stating the problem:** We have the differential equation $$\frac{dy}{dx} = -\frac{x}{1+x^2} y$$ and want to separate variables. 2. **Rewrite the equation:** The equation is $$

1. **State the problem:** Solve the differential equation $$(1 + x^2) \frac{dy}{dx} + xy = 0$$ with the initial condition $y(0) = 2$ using the method of separation of variables. 2.

1. **Problem Statement:** (i) Use the convolution theorem to find the inverse Laplace transform of $$\frac{1}{s(s^2 - 4)}$$ and show it equals $$\frac{1}{4}(\cosh 2t - 1)$$.

1. **State the problem:** Solve the differential equation $$(D^2 + a^2)y = \sec(ax)$$ using the method of variation of parameters. 2. **Identify the homogeneous equation:** The ass

1. **State the problem:** Solve the differential equation $$ (D^2 - 3D + 2)y = 2\cos(2x+3) + 2e^x $$ where $D = \frac{d}{dx}$. 2. **Identify the type of equation:** This is a nonho

1. **State the problem:** Solve the differential equation $$(1 + y^4) \frac{dy}{dx} = x^3 e^{x^2}$$ and identify the correct general solution from the given options. 2. **Rewrite t

1. The problem is to eliminate the constants $C_1$, $C_2$, and $C_3$ from the function $$y = C_1 e^{2x} + C_2 x e^{2x} + C_3 x^2 e^{2x}$$

1. **Stating the problem:** Solve the differential equation given by $$(9D^2 - 6D + 1)y = 0,$$ where $D$ represents the differential operator $\frac{d}{dx}$. 2. **Understanding the

1. **State the problem:** Solve the differential equation $$(2D^2 - 5D - 3)y = 0$$ where $D$ represents the differential operator $\frac{d}{dx}$. 2. **Rewrite the equation:** The o

1. **State the problem:** Solve the differential equation $$y'' = -y$$. 2. **Recall the formula and rules:** This is a second-order linear differential equation with constant coeff

1. **Problem Statement:** Solve the differential equation solvable for $p=\frac{dy}{dx}$: $$x\left(\frac{dy}{dx}\right)^2 + (y - 1 - x^2) \frac{dy}{dx} - x(y - 1) = 0$$

1. **Problem:** Solve the ODE $$(x^2 - 4) y' = -2xy - 6x$$ 2. **Rewrite the equation:**

1. The problem is to solve the differential equation $y'' + 3y' + 2y = 6$ using the method of undetermined coefficients. 2. The general approach is to find the complementary soluti

1. **State the problem:** Solve the differential equation $$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = \sin(e^x)$$ using variation of parameters. 2. **Find the homogeneous solution:

1. **Menyatakan masalah:** Kita ingin menentukan kapan Oetzi, mumi dari zaman Neolitikum, hidup dan meninggal berdasarkan rasio karbon radioaktif $^{14}_6C$ terhadap karbon biasa $

1. The problem states that we have a differential equation given by $$y'(x) = \frac{x}{3} + \frac{y}{6}$$ and a slope field representing this equation. 2. This is a first-order lin

1. The problem involves extending the slope-field tick mark from $x=0$ to $x=0.25$ and identifying the $y$-coordinate of the newest dot at $x=0.25$. 2. A slope field represents the