Title: Editing Openai/69118e47-8450-8000-9764-f97042c547df - freem URL Source: https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit Published Time: Fri, 16 Oct 2026 08:16:28 GMT Markdown Content: [Jump to content](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit#bodyContent) - [x] Main menu Main menu move to sidebar hide Navigation * [Main page](https://freemwiki.com/wiki/Main_Page "Visit the main page [alt-shift-z]") * [Recent changes](https://freemwiki.com/wiki/Special:RecentChanges "A list of recent changes in the wiki [alt-shift-r]") * [Random page](https://freemwiki.com/wiki/Special:Random "Load a random page [alt-shift-x]") * [Help about MediaWiki](https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents) Legal * [Terms](https://freemwiki.com/wiki/Terms_of_Service) * [Privacy Policy](https://freemwiki.com/wiki/Privacy_Policy) * [About](https://freemwiki.com/wiki/About) * [Contact](https://freemwiki.com/wiki/Contact) [![Image 1](https://freemwiki.com/images/freem.svg)**freem**](https://freemwiki.com/wiki/Main_Page) [Search](https://freemwiki.com/wiki/Special:Search "Search freem [alt-shift-f]") Search - 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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&veaction=edit§ion=4 "Edit section: gpt-5") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=4 "Edit section's source code: 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&veaction=edit§ion=5 "Edit section: User: Generate an image of a whiteboard with this solution worked out.") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=5 "Edit section's source code: 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&veaction=edit§ion=6 "Edit section: Assistant: Here’s your whiteboard-style solution!") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=6 "Edit section's source code: 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&veaction=edit§ion=7 "Edit section: User: Make a more photorealistic whiteboard image with markers and dusters lying around and signs of previous usage etc.") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=7 "Edit section's source code: 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&veaction=edit§ion=8 "Edit section: Assistant: Here you go — a more photorealistic whiteboard with markers, eraser, and light signs of previous usage is now generated.") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=8 "Edit section's source code: 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&veaction=edit§ion=9 "Edit section: User: Yes make it more messier with shadows and reflections and a wider shot as well.") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=9 "Edit section's source code: 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&veaction=edit§ion=10 "Edit section: Assistant: Done — here is a wider, messier, more photorealistic whiteboard scene with shadows, reflections, smudges, and markers/e…") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=10 "Edit section's source code: 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&veaction=edit§ion=11 "Edit section: References") | [edit source](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&action=edit§ion=11 "Edit section's source code: References")] 1. [↑](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit#cite_ref-1 "Jump up")["Power solution method"](https://chatgpt.com/share/69118e47-8450-8000-9764-f97042c547df). ChatGPT. Retrieved 2025-11-11. Retrieved from "[https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&oldid=592956](https://freemwiki.com/index.php?title=Openai/69118e47-8450-8000-9764-f97042c547df&oldid=592956)" ![Image 2](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit)[DISPLAYTITLE:Power solution method](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit)![Image 3](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit) _This conversation was exported from ChatGPT_[[1]](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit#)![Image 4](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit) ## gpt-5-thinking ### User: ![Image 5](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit) ![Image 6](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit) ### 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 ### 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! 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. 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. 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. 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… 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 1. [↑](https://freemwiki.com/index.php?section=2&title=Openai%2F69118e47-8450-8000-9764-f97042c547df&veaction=edit)["Power solution method"](https://chatgpt.com/share/69118e47-8450-8000-9764-f97042c547df). ChatGPT. Retrieved 2025-11-11. 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