Thread: Solutions curves

1. Solutions curves

Consider:
e^(-y)(dy/dt)+2 cost t=0

a) Solve it
I did this and got -e^(-y)=-2sint +C, or y=-ln(2sin t)+C

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

I'm not sure about this....I don't think so, since we have a negative ln....this one doesn't make too much sense to me though since 2 sin t will always be between -2 and 2, hence the only values -ln(2 sin t) can take are between -ln(-2) (which is not real) and -ln(2)...or I guess more appropriately -ln(1) and -ln(2)....

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

Probably not, but not sure.
Thanks!

2. Originally Posted by zhupolongjoe
Consider:
e^(-y)(dy/dt)+2 cost t=0

a) Solve it
I did this and got -e^(-y)=-2sint +C, or $\text{y=-ln(2sin t}$ $\color{red}\text{)+C}$

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

I'm not sure about this....I don't think so, since we have a negative ln....this one doesn't make too much sense to me though since 2 sin t will always be between -2 and 2, hence the only values -ln(2 sin t) can take are between -ln(-2) (which is not real) and -ln(2)...or I guess more appropriately -ln(1) and -ln(2)....

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

Probably not, but not sure.
Thanks!
$e^{-y}\frac{dy}{dt} = -2\cos(t) \implies \int e^{-y}\frac{dy}{dt} dt = -\int \cos(t)dt\implies -e^{-y} = -2\sin(t)+C$ by the reverse chain rule.
Therefore $y=-\ln(2\sin(t)+C)$. (Note how $C$ is inside the logarithm.)

For $y$ to exist, $-2\sin(t)+C>0$ for all $t$, otherwise the logarithm isn't defined.
Thus $C > 2$ yields a solution to exist for all real numbers since $-2\sin(t)+C>0$ for all $t$.
$-2 < C \leq 2$ yields a solution for some but not all real numbers since $-2\sin(t)+C>0$ for some $t$.
$C \leq -2$ yields no solution since $-2\sin(t)+C>0$ for no $t$.

Note, I'm assuming we're strictly working in the real numbers here.

3. Hi. I get $y(t)=-\log(2\sin(t)+c)$ and try and keep in mind that all differential equations are equally valid in the complex plane wherever the solution is analytic (differentiable in a region). So that solution above could just as well have been written with all complex variables like:

$w(z)=-\log(2\sin(z)+z_0)$, everything complex, and it would still satisfy the DE in some region of analyticity where the argument to the log function is not zero so c can't be 0 when t=0. When we solve for a "real" answer, we're just taking part of the "complex" solution that gives a "real" answer but we could just as well let c=-10 and would obtain the solution:

$y(t)=-\log(2\sin(t)-10)$ which gives $y(0)=-(\ln(10)-\pi i)$ and that is just as valid a solution as the real solution (in some region for which it is analytic).