# Eigenvectors of first-order differential equation

1. May 17, 2009

### jeff1evesque

Suppose that f is an eigenvector of T with corresponding eigenvalue $$\lambda$$. Then f' = T(f) = $$\lambda$$f. This is a first-order differential equation whose solutions are of the form f(t) = ce^($$\lambda$$*t) for some scalar constant c. Consequently, every real number $$\lambda$$ is an eigenvalue of T, and $$\lambda$$ corresponds to eigenvectors of the form ce^($$\lambda$$*t ) for c not equal to 0. Note that for $$\lambda$$ = 0, the eigenvectors are the nonzero constant functions.

Question: I was wondering if someone could explain the following sentence from above: "This is a first-order differential equation whose solutions are of the form f(t) = ce^($$\lambda$$ * t) for some scalar constant c. "

Work:
f ' = T(F) = $$\lambda$$f.
Therefore, f ' = $$\lambda$$ f ==> $$f' / f$$ = $$\lambda$$
So, [Integral] $$f ' / f$$ dt = [Integral] $$\lambda$$ dt
Therefore, ln(f) = $$\lambda$$t + c
<==> e^( ln(f) ) = e^( $$\lambda$$t + c )
= $$e^l^n^|^f^|$$ = e ^ ( $$\lambda$$t + c )
= $$e^c$$e^( $$\lambda$$ * t )
<==> f = $$e^c$$e^( $$\lambda$$ * t ) which does not equal ce^( lambda * t ) ??

Question: I'm kind of rusty with calculus but I didn't think [Integral] $$f ' / f$$ dt = ln(f)
I thought it would be [Integral] $$f ' / f$$ dt = [Integral] $$1 / f$$ dt = $$t / f$$

Last edited: May 17, 2009
2. May 18, 2009

### tiny-tim

Hi jeff1evesque!

(have a lambda: λ and an integral: ∫ and try using the X2 tag just above the Reply box )
Yes it does … the constant ec has simply been renamed c
∫f/'f dt = ln(f) + c because d/dt (ln(t)) = 1/t, so by the chain rule d/dt (ln(f(t))) = f'(t)/f(t)

3. May 18, 2009

### jeff1evesque

Wow, thanks a lot Tim.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9606107473373413, "perplexity": 1964.1264730868756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648003.58/warc/CC-MAIN-20180322205902-20180322225902-00036.warc.gz"}