Slope Field Solution Animation

Overview

A concise 15‑second visual demonstration of solving the first‑order linear ODE

$$\frac{dy}{dx}+2y=e^{-x}$$

The animation presents a clean white equation centered at the top with generous vertical padding, a dense blue‑dash slope field filling the remainder of the frame, and a glowing red solution curve that smoothly traces the field.

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Phases

#	Phase Name	Duration	Description
1	Grid & Field	~4 s	A light‑gray 2‑D coordinate grid fades in, occupying the lower ~80 % of the frame (below the top padding). The plane is populated with a dense field of short blue dashes representing the direction field $\vec v(x,y)=(1,\,e^{-x}-2y)$.
2	Equation Display	~3 s	The equation $\displaystyle \frac{dy}{dx}+2y=e^{-x}$ appears instantly (no slide) in white, centered horizontally at the top of the frame with noticeable vertical padding separating it from the grid.
3	Integrating Factor	~3 s	The integrating factor $e^{2x}$ slides in from the right, aligns next to the white equation, and the left‑hand side transforms into $\displaystyle \frac{d}{dx}\!\bigl(e^{2x}y\bigr)$ while the right‑hand side becomes $e^{2x}e^{-x}=e^{x}$.
4	Solution Curve	~5 s	A glowing red curve representing the solution

$$y(x)=Ce^{-2x}+\tfrac13 e^{-x}$$

(with a chosen constant for visual clarity) is drawn smoothly across the dense dash field, following the direction indicated by the dashes. |

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                TOP‑CENTER EQUATION          │
│            (white, with vertical padding)   │
│                                             │
│─────────────────────────────────────────────│
│                                             │
│            MAIN (visual) – GRID & FIELD      │
│   Light‑gray grid, dense blue dashes, red   │
│   glowing solution curve                    │
│                                             │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top‑Center Equation	White equation $\frac{dy}{dx}+2y=e^{-x}$ (static) with significant vertical padding from the grid.	Positioned at the very top, centered horizontally; no motion other than the integrating‑factor addition in Phase 3.
Main	Light‑gray coordinate grid, dense blue slope dashes, integrating‑factor animation, glowing red solution curve.	Grid fills the remainder of the frame beneath the equation; all visual elements are centered and use a minimalist vector style.
Caption	None – no persistent footer required.	Omitted to keep the design minimal.

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Notes

• Duration: Total ~15 seconds, matching the requested length.
• Style: Minimalist vector graphics; blue dashes for the field, glowing red for the solution, white for the equation, light‑gray grid. All motions are smooth and continuous.
• Assumptions:
  1. The slope field is defined by $\displaystyle \frac{dy}{dx}=e^{-x}-2y$.
  2. The integrating factor is $e^{2x}$ and the combined derivative is $\displaystyle \frac{d}{dx}\!\bigl(e^{2x}y\bigr)$.
  3. A specific constant $C$ (e.g., $C=1$) is chosen to produce a clearly visible curve.
• Only one Manim Scene class is required; phases occur sequentially within that scene.
• No additional text beyond the equation and integrating factor is displayed, preserving the clean, minimalist aesthetic.