I've got the AP Calculus AB exam tomorrow and I'm trying to study. There is a concept I don't get. It says "Consider the differential equation dy/dx=x/y where y does not equal 0." It gives me a slope field and I get that part of it, but then it asks me to write an equation for the line tangent to the solution curve that passes through the point (1,2). There are other questions like this too. Any help with how to do this so that I can figure out the concept would be beyond appreciated.

Question

amistre64 · Answer

you have a slope definition:  dy/dx defines the slope of a line at any point in the plane

amistre64 · Answer

dy/dx = 1/2 for the point (1,2)  therefore the slope of a line is 1/2  and the point its attached to is (1,2)

how do we make a line from this?

amistre64 · Answer

we can use the slope formula, let (1,2) and (x,y) be 2 points in the plane, and the slope between them is 1/2

$$\frac{y-2}{x-1}=\frac12$$

amistre64 · Answer

solve it for whatever form of a line equation you like best

athenawolf · Answer

Thank you so much! I can't believe I wasn't getting that before. In hindsight it seems blatantly obvious. :)

amistre64 · Answer

:)  yeah, it does lol