Question

Help please!
How would I sketch an integral curve for dy/dx = x/y?

Answer 1

what is your solution to this differential equation?

Answer 2

dy/dx=x/y

y*dy=x*dx

Answer 3

y = x^2/2 ?

Answer 4

@sarahseburn not quite\[\frac{dy}{dx}=\frac xy\]\[ydy=xdx\]\[\int ydy=\int xdx\]careful now, what is the result of integrating both sides

Answer 5

y=x ?

Answer 6

\[\int ydy=y~~~???\]I don't think so\[\int ydy=\frac12y^2+C\](btw, in differential equations remembering that constant of integration is \(critical\))
if you are taking DE's you need to be more careful than that, but I am actually messing this up myself, we don't even need to solve the DE to get the integral curves...

Answer 7

\[y'=x/y\]now let's look for some interesting possible values of y'
under which conditions will y'=1 for example?

Answer 8

ok .. so we dont need to solve for the intergral curve then? i didn't think so .. it's just supposed to be a sketch

Answer 9

y' =1 when y=x

Answer 10

right, so we want to know what the derivative will be for each point (x,y)
we obviously can't draw them all, so start by looking for some particular cases

...yes y'=1 when y=x, that mean that for all points along the line y=x we have a derivative with a slope of 1

Answer 11

to illustrate the derivative at various points we draw little arrows...

Answer 12

so that is one integral curve
what about another condition; when will y'=0 ?

Answer 13

y'=0 when x=0 ?

Answer 14

right, so all along the line x=0 (i.e. the y-axis) we will have slope 0....

Answer 15

make sense?

Answer 16

kind of yes ... so will the integral curve be a circle?

Answer 17

curves *

Answer 18

circle? no where do you see a circle coming from?
no let's look at a few more things; what happens to y' as x increases?

Answer 19

...if y is kept constant that is...

Answer 20

it increases
soo .. the curves will just be curving up then right?

Answer 21

exactly

Answer 22

let's look along the line y=1 as x increases...
at x=2, y'=2
at x=3, y'=3, etc

Answer 23

ok.

Answer 24

as \(x\to\infty,y'\to\infty\) so the integral curve will curve sharply upward along y=1