Thread: Slope Fields/Differential Equations

a) Find the general solution of:

e^(-y)*(dy/dt)+2 cos t=0.

I'm quite sure this is y=-ln(2 sin(t)+C)

b) Does every value of C correspond to a solution curve? If not, which Cs occur and which do not?

Is it just that you need 2 sin(t)+C>0 and so it occurs if C>-2 sin (t)? Or is there something more I can say here? Obviously no C will exist that is less than -2 and also all Cs will exist if C is greater than 2, but between -2 and 2 it depends on the value of t?

c) Do all solutions have the same domain? Explain.

Again, I'd say no. For instance -ln(2 sin(t)+2) has domain sin(t)>-1 and -ln (2 sin (t)+3) has domain sin(t)>-3/2, but actually this is the same. I am not quite sure how to explain this one, but quite sure it is NO.

Originally Posted by zhupolongjoe

a) Find the general solution of:

e^(-y)*(dy/dt)+2 cos t=0.

I'm quite sure this is y=-ln(2 sin(t)+C)

b) Does every value of C correspond to a solution curve? If not, which Cs occur and which do not?

Is it just that you need 2 sin(t)+C>0 and so it occurs if C>-2 sin (t)? Or is there something more I can say here? Obviously no C will exist that is less than -2 and also all Cs will exist if C is greater than 2, but between -2 and 2 it depends on the value of t?

c) Do all solutions have the same domain? Explain.

Again, I'd say no. For instance -ln(2 sin(t)+2) has domain sin(t)>-1 and -ln (2 sin (t)+3) has domain sin(t)>-3/2, but actually this is the same. I am not quite sure how to explain this one, but quite sure it is NO.

Your solution for a) is correct.

For b) you need to analyze three different cases



As you said the first cases has not real solutions. The 2nd case will exists for some but not all and the last cases will exists for all t.

This will help you with part C as well.