Question

Fun Fun Fun. Find the general solution of the given differential equation.Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. cos(x)y'+(sin(x))y=1

Answer 1

\[e^{\int\limits{}_{}p(x)dx}=e^{\int\limits{}_{}\tan(x)dx}\]

Answer 2

=\[e^{-\ln(\cos(x)+c}\]

Answer 3

=\[=e^{\ln(\cos(x^{-1})+c}\]

Answer 4

\[=\cos(x^{-1})+c\]

Answer 5

\[\int\limits_{}^{}\frac{d}{dx}[\cos(x^{-1})y]=\int\limits_{}^{}\frac{1}{\cos(x)}*\cos(x^{-1})dx\]

Answer 6

\[\cos(x^{-1})y=\int\limits{}_{}\frac{\cos(x^{-1}}{\cos(x)}dx\]

Answer 7

\[e ^{\int\limits \tan(x)} = e ^{-\ln \cos(x)} = e ^{\ln {1 \over \cos(x)}} = \sec(x)\]

Answer 8

aha not simplified enough alright

Answer 9

\[\sec(x)y=\int\limits_{}^{}\sec^2(x)dx\]

Answer 10

\[\sec(x)y=\tan(x)+c\]

Answer 11

so far soo good?

Answer 12

yeah the book just multiplies by cos(x) to get

y=sin(x)+c(cos(x)

book makes no sense at all -.- leaves it in c= and then sometimes y=

Answer 13

because sec(x) = 1/cos(x), they multiply both sides by cos(x) to isolate y

Answer 14

but yes, looks good Outkast3r09

Answer 15

Looks good to me. This one is easy to solve for y and its usually best to have an explicit solution

Answer 16

yeah i undrstand that but sometimes it will say differential general and still leave it as c=

anyways i was away for cases could you expain how this works jim"?

Answer 17

not sure what you mean

Answer 18

well in this equation ther eis only one case

but sometimes you can get two cases such as

case 1: x cannot equal 0 case 2: x = 0

Answer 19

it has to do when you divide by something just didn't understand it the way he explained i'm sure i'd understand if he had explained it

Answer 20

hmm well I see no domain issues with y=sin(x)+c*cos(x), so you don't have to worry about cases there...but...you may have to worry about cases in intermediate steps. I'm not too sure.

Answer 21

trying to find a case problem hold on

Answer 22

alright, I'm looking up dealing with cases, but not finding much

Answer 23

yeah the book doesn't explain it as much but my teacher is a Dr in Differential Equations and always checks every case.. most of the time it's just the general but i've seen problems where there was 2 answers

Answer 24

hmm alright, let me look some more

Answer 25

You might be on to something when you say y(0)

So y(0)=sin(0)+c*cos(0) ---> y(0) = c

And for nonzero x, y(x)=sin(x)+c*cos(x)

Another case is x = kpi/2 where k is some integer. If this is the case, then

y(x) = (-1)^(k+1)


Not sure if I thought of every possible case though.