Editing Openai/69118e47-8450-8000-9764-f97042c547df


{{DISPLAYTITLE:Power solution method}}

''This conversation was exported from ChatGPT''<ref>{{cite web|title=Power solution method|url=https://chatgpt.com/share/69118e47-8450-8000-9764-f97042c547df|publisher=ChatGPT|access-date=2025-11-11}}</ref>

== gpt-5-thinking ==
=== User:  ===


=== Assistant: Try a power ypCx4yp  Cx^4ypCx4 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 ==
<references />


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