Find an equation of the curve that satisfies

shayne704 said:
dy/dx=108yx^17
and whos y-intercept is 6
y(x)=___________________
Click to expand...

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.
 
shayne704 said:
dy/dx=108yx^17
and whos y-intercept is 6
y(x)=___________________
Click to expand...

Separate the variables: dy/y = 108x^17

Now take the integrals of both sides.
 
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.
 
Ok, shayne, I'll do this one for you.

Given: dydx = 108yx17and y(0) = 6, find y(x).\displaystyle Given: \ \frac{dy}{dx} \ = \ 108yx^{17}and \ y(0) \ = \ 6, \ find \ y(x).

First note that y = 0 is also a solution. To find other solutions, we assume y ≠0 and we\displaystyle First \ note \ that \ y \ = \ 0 \ is \ also \ a \ solution. \ To \ find \ other \ solutions, \ we \ assume \ y \ \ne0 \ and \ we

separate variables as follows.\displaystyle separate \ variables \ as \ follows.

∫dyy = ∫108x17dx\displaystyle \int\frac{dy}{y} \ = \ \int108x^{17}dx

ln∣y∣ = 108x1818+C = 6x18+C\displaystyle ln|y| \ = \ \frac {108x^{18}}{18}+C \ = \ 6x^{18}+C

∣y∣ = ±eCe6x18\displaystyle |y| \ = \ \pm e^{C} e^{6x^{18}}

y(x) = Ae6x18 (general solution), A = ±eC, A = 0.\displaystyle y(x) \ = \ Ae^{6x^{18}} \ (general \ solution), \ A \ = \ \pm e^{C}, \ A \ = \ 0.

y(0) = 6 = A(1), hence y(x) = 6e6x18\displaystyle y(0) \ = \ 6 \ = \ A(1), \ hence \ y(x) \ = \ 6e^{6x^{18}}

Check: y′(x) = 6[(18)(6)x17]e6x18, and dydx = 108(6e6x18)x17, y′(x) =dydx, QED\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

Also y(x) = 0, then y′(x) =0 = 108(0)x17 = dydx\displaystyle Also \ y(x) \ = \ 0, \ then \ y'(x) \ =0 \ = \ 108(0)x^{17} \ = \ \frac{dy}{dx}
 
After thought: Given:dydx = 108yx17 and y(0) = 6\displaystyle Given: \frac{dy}{dx} \ = \ 108yx^{17} \ and \ y(0) \ = \ 6

y = 0 is also a solution. Why?\displaystyle y \ = \ 0 \ is \ also \ a \ solution. \ Why?

If y = 0, then dydx = 0,and the only way this can happen is if y = 0, because if\displaystyle If \ y \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, and \ the \ only \ way \ this \ can \ happen \ is \ if \ y \ = \ 0, \ because \ if

 x = 0, dydx will also equal zero, but y will equal 6.\displaystyle \ x \ = \ 0, \ \frac{dy}{dx} \ will \ also \ equal \ zero, \ but \ y \ will \ equal \ 6.

If x = 0, then dydx = 0, but y = 6.\displaystyle If \ x \ = \ 0, \ then \ \frac{dy}{dx} \ = \ 0, \ but \ y \ = \ 6.

Look at it this way, if dydx = 0, y may or may not equal 0, for example let y = 5, dydx = 0;\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;

 however if y =0, dydx must = 0.\displaystyle \ however \ if \ y \ =0, \ \frac{dy}{dx} \ must \ = \ 0.

In other words, y(x) = 6e6x18, (6e6x18 is always > than 0),but we must remember\displaystyle In \ other \ words, \ y(x) \ = \ 6e^{6x^{18}}, \ (6e^{6x^{18}} \ is \ always \ > \ than \ 0), but \ we \ must \ remember

 in the back of our minds that y(x) = 0 is also a solution.\displaystyle \ in \ the \ back \ of \ our \ minds \ that \ y(x) \ = \ 0 \ is \ also \ a \ solution.

y(x) = 6e6x18,only when we assume that y ≠ 0.\displaystyle y(x) \ = \ 6e^{6x^{18}}, only \ when \ we \ assume \ that \ y \ \ne \ 0.
 
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