# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-1 => Topic started by: Victor Ivrii on September 28, 2018, 03:30:24 PM

Title: Q1: TUT0301
Post by: Victor Ivrii on September 28, 2018, 03:30:24 PM
Find the general solution of the given differential equation, and use it to determine how solutions behave as $t\to\infty$.
\begin{equation*}
y' - 2y = e^{2t},\qquad y(0)=2.
\end{equation*}
Title: Re: Q1: TUT0301
Post by: Jiabei Bi on September 28, 2018, 06:00:22 PM
$$y' - 2y = e^{2t}\\\mu (x) = e^{\int-2\ dt} = e^{-2t}\\\frac{d}{dt} (e^{-2t}y) = e^{-2t}y' -2e^{-2t} y = e^{-2t}e^{2t}= 1\\ \text{integral on both side gives:}\\ e^{-2t}y = t + C\\y = te^{2t} + C{e^{2t}}\\y(0) = 0 + C = 2 \implies C = 2\\\text{thus} \ \ y =e^{2t}(t+2)\\\text{and} \ \ t \to \infty \implies y \to \infty$$
{"ft_lang_label":"__label__en","ft_lang_prob":0.6285,"math_prob":0.9998573,"size":667,"snap":"2022-40-2023-06","text_gpt3_token_len":295,"char_repetition_ratio":0.12971343,"word_repetition_ratio":0.0,"special_character_ratio":0.49025488,"punctuation_ratio":0.072463766,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991096,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-03T20:30:34Z\",\"WARC-Record-ID\":\"<urn:uuid:3209efb1-3f49-4b8f-a3a7-97a7dcb2bc0a>\",\"Content-Length\":\"4935\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5196bd17-9c4d-4716-9a2e-4befa0e07169>\",\"WARC-Concurrent-To\":\"<urn:uuid:dfc3b448-8146-4886-aa7b-9e1e3ce9502f>\",\"WARC-IP-Address\":\"142.150.233.214\",\"WARC-Target-URI\":\"https://forum.math.toronto.edu/index.php?PHPSESSID=m7ol7vsqn1khd034i0rsgdrd25&action=printpage;topic=1272.0\",\"WARC-Payload-Digest\":\"sha1:EIGVTD63ZV6SWCRUQAMZ2F67J7OWMCT5\",\"WARC-Block-Digest\":\"sha1:KUPIRD3WMKYVOCAHOPWPN6KSGYFSH54G\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337432.78_warc_CC-MAIN-20221003200326-20221003230326-00337.warc.gz\"}"}