Question

solve the differential equation: y''+y=1/sinx

Answer 1

differential equation \(\implies \) lalaly

Answer 2

first find the homogeneous solution
let\[y''+y=0\]so\[r^2+1=0\]\[r^2=-1\]\[r= \pm i\]so\[y_h=c_1cosx+c_2sinx\]

Answer 3

now to find the particular solution we have two methods, which one do u like to use? lol

Answer 4

When u decide let me know :D:D im here

Answer 5

thanks ! i got the homogenous solution but to which methods do you mean ? i tried variation of parameters but got stucked with crazy integral

Answer 6

theres method of undetermined coeffecients

Answer 7

i don't know how to do it because its 1/sinx and not just sinx, if you can i will be happy if you show me the two ways =]

Answer 8

ok can u give me sometime i need to do it on my own first then i can write it down here

Answer 9

great !! thanks alot !!

Answer 10

ok variation of parameters
first find the wronskian W
\[y_1=cosx\]\[y_2=sinx\]
let \[g(x)=\frac{1}{sinx}\]so the particular solution is\[y_p=-y_1 \int\limits{\frac{y_2 \times g(x)}{W}dx}+y_2 \int\limits{\frac{y_1 \times g(x)}{W}dx}\]\[y_p=-cosx \int\limits{\frac{sinx \times \frac{1}{sinx}}{1}dx}+sinx \int\limits{\frac{cosx \times \frac{1}{sinx}}{1}dx}\]