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Find an equation of the curve that satisfies
- Thread starter shayne704
- Start date
D
Deleted member 4993
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shayne704 said:dy/dx=108yx^17
and whos y-intercept is 6
y(x)=___________________
You posted three different (rather straight-forward) problems - without a line of work.
Looks like a take-home test...
Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.
fasteddie65
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shayne704 said:dy/dx=108yx^17
and whos y-intercept is 6
y(x)=___________________
Separate the variables: dy/y = 108x^17
Now take the integrals of both sides.
BigGlenntheHeavy
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shayne704, just exactly, what is your problem?
I concur wholeheartedly with Subhotosh Khan in regards to your postings.
I don't mind doing a problem to conclusion when I think it might be of aid to a fellow aspirant (not to mentioned my ego),
however, when I think I'm being "had", forget it.
I concur wholeheartedly with Subhotosh Khan in regards to your postings.
I don't mind doing a problem to conclusion when I think it might be of aid to a fellow aspirant (not to mentioned my ego),
however, when I think I'm being "had", forget it.
D
Deleted member 4993
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BigGlenntheHeavy
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Ok, shayne, I'll do this one for you.
\(\displaystyle Given: \ \frac{dy}{dx} \ = \ 108yx^{17}and \ y(0) \ = \ 6, \ find \ y(x).\)
\(\displaystyle First \ note \ that \ y \ = \ 0 \ is \ also \ a \ solution. \ To \ find \ other \ solutions, \ we \ assume \ y \ \ne0 \ and \ we\)
\(\displaystyle separate \ variables \ as \ follows.\)
\(\displaystyle \int\frac{dy}{y} \ = \ \int108x^{17}dx\)
\(\displaystyle ln|y| \ = \ \frac {108x^{18}}{18}+C \ = \ 6x^{18}+C\)
\(\displaystyle |y| \ = \ \pm e^{C} e^{6x^{18}}\)
\(\displaystyle y(x) \ = \ Ae^{6x^{18}} \ (general \ solution), \ A \ = \ \pm e^{C}, \ A \ = \ 0.\)
\(\displaystyle y(0) \ = \ 6 \ = \ A(1), \ hence \ y(x) \ = \ 6e^{6x^{18}}\)
\(\displaystyle Check: \ y'(x) \ = \ 6[(18)(6)x^{17}]e^{6x^{18}}, \ and \ \frac{dy}{dx} \ = \ 108(6e^{6x^{18}})x^{17}, \ y'(x) \ = \frac{dy}{dx}, \ QED\)
\(\displaystyle Also \ y(x) \ = \ 0, \ then \ y'(x) \ =0 \ = \ 108(0)x^{17} \ = \ \frac{dy}{dx}\)
\(\displaystyle Given: \ \frac{dy}{dx} \ = \ 108yx^{17}and \ y(0) \ = \ 6, \ find \ y(x).\)
\(\displaystyle First \ note \ that \ y \ = \ 0 \ is \ also \ a \ solution. \ To \ find \ other \ solutions, \ we \ assume \ y \ \ne0 \ and \ we\)
\(\displaystyle separate \ variables \ as \ follows.\)
\(\displaystyle \int\frac{dy}{y} \ = \ \int108x^{17}dx\)
\(\displaystyle ln|y| \ = \ \frac {108x^{18}}{18}+C \ = \ 6x^{18}+C\)
\(\displaystyle |y| \ = \ \pm e^{C} e^{6x^{18}}\)
\(\displaystyle y(x) \ = \ Ae^{6x^{18}} \ (general \ solution), \ A \ = \ \pm e^{C}, \ A \ = \ 0.\)
\(\displaystyle y(0) \ = \ 6 \ = \ A(1), \ hence \ y(x) \ = \ 6e^{6x^{18}}\)
\(\displaystyle Check: \ y'(x) \ = \ 6[(18)(6)x^{17}]e^{6x^{18}}, \ and \ \frac{dy}{dx} \ = \ 108(6e^{6x^{18}})x^{17}, \ y'(x) \ = \frac{dy}{dx}, \ QED\)
\(\displaystyle Also \ y(x) \ = \ 0, \ then \ y'(x) \ =0 \ = \ 108(0)x^{17} \ = \ \frac{dy}{dx}\)
BigGlenntheHeavy
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After thought: \(\displaystyle Given: \frac{dy}{dx} \ = \ 108yx^{17} \ and \ y(0) \ = \ 6\)
\(\displaystyle y \ = \ 0 \ is \ also \ a \ solution. \ Why?\)
\(\displaystyle If \ y \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, and \ the \ only \ way \ this \ can \ happen \ is \ if \ y \ = \ 0, \ because \ if\)
\(\displaystyle \ x \ = \ 0, \ \frac{dy}{dx} \ will \ also \ equal \ zero, \ but \ y \ will \ equal \ 6.\)
\(\displaystyle If \ x \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, \ but \ y \ = \ 6.\)
\(\displaystyle Look \ at \ it \ this \ way, \ if \ \frac{dy}{dx} \ = \ 0, \ y \ may \ or \ may \ not \ equal \ 0, \ for \ example \ let \ y \ = \ 5, \ \frac{dy}{dx} \ = \ 0;\)
\(\displaystyle \ however \ if \ y \ =0, \ \frac{dy}{dx} \ must \ = \ 0.\)
\(\displaystyle In \ other \ words, \ y(x) \ = \ 6e^{6x^{18}}, \ (6e^{6x^{18}} \ is \ always \ > \ than \ 0), but \ we \ must \ remember\)
\(\displaystyle \ in \ the \ back \ of \ our \ minds \ that \ y(x) \ = \ 0 \ is \ also \ a \ solution.\)
\(\displaystyle y(x) \ = \ 6e^{6x^{18}}, only \ when \ we \ assume \ that \ y \ \ne \ 0.\)
\(\displaystyle y \ = \ 0 \ is \ also \ a \ solution. \ Why?\)
\(\displaystyle If \ y \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, and \ the \ only \ way \ this \ can \ happen \ is \ if \ y \ = \ 0, \ because \ if\)
\(\displaystyle \ x \ = \ 0, \ \frac{dy}{dx} \ will \ also \ equal \ zero, \ but \ y \ will \ equal \ 6.\)
\(\displaystyle If \ x \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, \ but \ y \ = \ 6.\)
\(\displaystyle Look \ at \ it \ this \ way, \ if \ \frac{dy}{dx} \ = \ 0, \ y \ may \ or \ may \ not \ equal \ 0, \ for \ example \ let \ y \ = \ 5, \ \frac{dy}{dx} \ = \ 0;\)
\(\displaystyle \ however \ if \ y \ =0, \ \frac{dy}{dx} \ must \ = \ 0.\)
\(\displaystyle In \ other \ words, \ y(x) \ = \ 6e^{6x^{18}}, \ (6e^{6x^{18}} \ is \ always \ > \ than \ 0), but \ we \ must \ remember\)
\(\displaystyle \ in \ the \ back \ of \ our \ minds \ that \ y(x) \ = \ 0 \ is \ also \ a \ solution.\)
\(\displaystyle y(x) \ = \ 6e^{6x^{18}}, only \ when \ we \ assume \ that \ y \ \ne \ 0.\)