Title: Openai/69118e47-8450-8000-9764-f97042c547df - freem

URL Source: https://freemwiki.com/index.php?mobileaction=toggle_view_mobile&title=Openai%2F69118e47-8450-8000-9764-f97042c547df

Published Time: Sun, 30 Aug 2026 03:02:39 GMT

Markdown Content:
_This conversation was exported from ChatGPT_[[1]](https://freemwiki.com/index.php?mobileaction=toggle_view_mobile&title=Openai%2F69118e47-8450-8000-9764-f97042c547df#cite_note-1)

## gpt-5-thinking[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=1 "Edit section: gpt-5-thinking")

### User:[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=2 "Edit section: User:")

### Assistant: Try a power ypCx4yp Cx^4yp​Cx4 since the ODE is Cauchy–Euler and r4r4r4 isn’t a root of the homogeneous part who…[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=3 "Edit section: Assistant: Try a power ypCx4yp  Cx^4yp​Cx4 since the ODE is Cauchy–Euler and r4r4r4 isn’t a root of the homogeneous part who…")

Try a power yp=Cx4y_p = Cx^4yp​=Cx4 (since the ODE is Cauchy–Euler and r=4r=4r=4 isn’t a root of the homogeneous part whose roots are 1,2,−11,2,-11,2,−1).

Compute

yp′=4Cx3,yp′′=12Cx2,yp′′′=24Cx.y_p'=4Cx^3,\quad y_p _=12Cx^2,\quad y\_p'_=24Cx.yp′​=4Cx3,yp′′​=12Cx2,yp′′′​=24Cx. Plug into

x3y′′′+x2y′′−2xy′+2y=2x4:x^3y'_+x^2y_-2xy'+2y=2x^4:x3y′′′+x2y′′−2xy′+2y=2x4: x3(24Cx)+x2(12Cx2)−2x(4Cx3)+2(Cx4)=(24+12−8+2)C x4=30C x4.x^3(24Cx)+x^2(12Cx^2)-2x(4Cx^3)+2(Cx^4) = (24+12-8+2)C\,x^4=30C\,x^4.x3(24Cx)+x2(12Cx2)−2x(4Cx3)+2(Cx4)=(24+12−8+2)Cx4=30Cx4. Set 30C x4=2x4⇒C=11530C\,x^4=2x^4\Rightarrow C=\tfrac{1}{15}30Cx4=2x4⇒C=151​.

yp(x)=x415 \boxed{\,y_p(x)=\frac{x^4}{15}\,}yp​(x)=15x4​​

## gpt-5[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=4 "Edit section: gpt-5")

### User: Generate an image of a whiteboard with this solution worked out.[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=5 "Edit section: User: Generate an image of a whiteboard with this solution worked out.")

Generate an image of a whiteboard with this solution worked out.

### Assistant: Here’s your whiteboard-style solution![edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=6 "Edit section: Assistant: Here’s your whiteboard-style solution!")

Here’s your whiteboard-style solution! If you'd like a cleaner layout, color markers, arrows, or a more “classroom” style board, just tell me — I can regenerate with any style you want.

(Image is already generated above.)

If you need the image downloaded, embedded into LaTeX, or turned into a PDF/PowerPoint, I can do that too.

### User: Make a more photorealistic whiteboard image with markers and dusters lying around and signs of previous usage etc.[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=7 "Edit section: User: Make a more photorealistic whiteboard image with markers and dusters lying around and signs of previous usage etc.")

Make a more photorealistic whiteboard image with markers and dusters lying around and signs of previous usage etc.

### Assistant: Here you go — a more photorealistic whiteboard with markers, eraser, and light signs of previous usage is now generated.[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=8 "Edit section: Assistant: Here you go — a more photorealistic whiteboard with markers, eraser, and light signs of previous usage is now generated.")

Here you go — a more photorealistic whiteboard with markers, eraser, and light signs of previous usage is now generated.

If you'd like:

✅ a messier board ✅ shadows / reflections / room background ✅ a wider shot of the whiteboard in a classroom ✅ different marker colors or handwriting

Just tell me and I’ll generate a new version.

### User: Yes make it more messier with shadows and reflections and a wider shot as well.[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=9 "Edit section: User: Yes make it more messier with shadows and reflections and a wider shot as well.")

Yes make it more messier with shadows and reflections and a wider shot as well.

### Assistant: Done — here is a wider, messier, more photorealistic whiteboard scene with shadows, reflections, smudges, and markers/e…[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=10 "Edit section: Assistant: Done — here is a wider, messier, more photorealistic whiteboard scene with shadows, reflections, smudges, and markers/e…")

Done — here is a wider, messier, more photorealistic whiteboard scene with shadows, reflections, smudges, and markers/erasers.

If you want:

✅ a classroom background ✅ overhead lighting reflections ✅ ghosted previous equations ✅ colored smudges / dry-erase dust ✅ tilted angle / natural camera shot

Just tell me — I can make it even more realistic.

## References[edit](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit&section=11 "Edit section: References")

1.   [↑](https://freemwiki.com/index.php?mobileaction=toggle_view_mobile&title=Openai%2F69118e47-8450-8000-9764-f97042c547df#cite_ref-1 "Jump up")["Power solution method"](https://chatgpt.com/share/69118e47-8450-8000-9764-f97042c547df). ChatGPT. Retrieved 2025-11-11.