11.2 Initial value problem in explicit form.

karush

Well-known member

Given #11
$\quad\displaystyle xdx+ye^{-x}dy=0,\quad y(0)=1$
a. Initial value problem in explicit form.
$\quad xdx=-ye^{-x}dy$
separate
$\quad \frac{x}{e^{-x}}\, dx=-y\, dy$
simplify
$\quad xe^x\, dx=-y\, dy$
rewrite
$\quad y\,dy=-xe^x\,dx$
integrate (with boundaries)
$\quad \int_1^y u\,dy=-\int_0^x ve^v\,dv$

OK i did this so far hopefully ok but didnt know how to do b and c (on desmos)

b. Plot the graph of the solution
c. Interval of solution.

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Cbarker1

Active member
Re: 11.2

I recommend that you integrate both side to find the curve and the graph as well as provide that answer to us.

HallsofIvy

Well-known member
MHB Math Helper
Re: 11.2

(You have a typo- your "dy" should be "du".)

Why stop there? Obviously you have to do the integrals! The "du" integral is straight forward. To do the "dv" integral, use "integration by parts".

Solving for y will involve a square root. The domain is determined by the fact that the square root must be of a non-negative number. The graph will have two parts, one with "+", the other with "-".

MarkFL

Staff member
Re: 11.2

What title would you like me to give this thread so that the title is useful?

karush

Well-known member
Re: 11.2

11.2 Initial value problem in explicit form.

I tried to fix it but couldn't

==================================

integrate (with boundaries)
$$\int_1^y y\,dy=-\int_0^x xe^x\,dx$$
then
$\frac{y^2}{2}-\frac{1}{2}=-e^xx+e^x-1$
isolate y
multiply thru by 2 and add 1
$y^2=-2e^xx+2e^x-2+1$
simplify RHS and factor
$y^2=2[1-x]e^x-1$
square both sides
$y = 2[(1-x)e^x-1]^{1/2}$

typo maybe??

when this is graphed you get nothing????

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topsquark

Well-known member
MHB Math Helper
Re: 11.2

thake the square (root) of both sides
$y = 2[(1-x)e^x-1]^{1/2}$

typo maybe??

when this is graphed you get nothing????
The "2" belongs under the square root. Typo?

Here's the graph. I have no idea why this would cause you trouble.

And don't forget the part of the solution with the negative sign!

-Dan

karush

Well-known member

here is the desmos attempt

also I don't know where the interval of $−1.68<x<0.77$ which is the book answer comes from
I set y=0 but that didn't return those values

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HallsofIvy

Well-known member
MHB Math Helper
here is the desmos attempt

also I don't know where the interval of $−1.68<x<0.77$ which is the book answer comes from
I set y=0 but that didn't return those values
What values did you get? If I graph the function on Desmos (https://www.desmos.com/calculator) clicking on the y-intercept, I get x= -1.678 and x= 0.768)

If y= 0 then $$2(1- x)e^x- 1= 0$$ then $$2(1- x)e^x= 1$$.

Let u= 1- x so that x= 1- u. The equation becomes $$2ue^{1- u}= 2eue^{-u}= 1$$. Let v= -u so that u= -v. The equation becomes $$-2eve^v= 1$$. Divide both sides by -2e. $$ve^v= -\frac{1}{2e}$$. Then $$v= W\left(\frac{-1}{2e}\right)$$ where "W" is "Lambert's W function, defined as the inverse function to $$f(x)= xe^x$$.