- #1

garr6120

- 42

- 0

## Homework Statement

I am having trouble proving if the equation i have found for number 1 is correct. I have posted my solution to get back to the main problem in the first photo below.

For number 2 I am having trouble isolating for 1 y(x). Did i do the integration and setup properly?

## Homework Equations

Question 1:

given y' + ycotanx+2cosx (1) and y(π/2)=0 find the IVP.

find the integrating factor from equation 1.

μ(x)=e

^{∫cotanx}dx=sinx (2)

multiply equation 2 by 1.

y'sinx+ysinxcotanx=2cosxsinx

(sinxy)'=2cosxsinx (3)

integrate equation 3.

sinxy=∫2cosxsinx=sin

^{2}x+c (4)

Isolate y in 4.

y=sinx+c(csc

^{2}x)

plug in the initial value to find c which is found to be -1.

y=sinx-csc

^{2}x (5)

proving that this is a solution to the differential:

take the derivative of equation 5.

y'=cosx+2csc

^{2}xcotanx

plug y and y' into differential equation 1.

y'+ycotanx=cosx+2csc

^{2}xcotanx+cosx-cotanxcsc

^{2}x

2cosx+3csc

^{2}xcotanx

I cannot get an answer for 2cosx. I did everything right.

For Question 2:

y'+ytanx=y

^{2}(1) for y(0)=1/2

find an integration factor.

μ(x)=e

^{∫tanx}dx=1/cosx {2}

multiply equation 1 by 2.

(y(x)/cosx)'=y

^{2}/cosx (3)

integrating equation 3.

y(x)/cosx=y

^{2}∫secx=y

^{2}ln|secx+tanx|+c (4)

isolating y in equation 4.

y(x)=cosxy

^{2}ln|secx+tanx|+c(cosx) (4)

Here is where I get stuck i don't know how to isolate for y.

## The Attempt at a Solution

.[/B]#### Attachments

Last edited: