1. ## First-Order Differential Equations

How would you solve this problem? Here is the question:

Given the differential equation: y' + y tanx= cosx

(a) Verify by subsitution that for any number C, y(x) = (x+C) cosx satisfies the given differential equation.
(b) Determine a value of the constant C so that y(3.14)= 0

2. Originally Posted by googoogaga
How would you solve this problem? Here is the question:

Given the differential equation: y' + y tanx= cosx

(a) Verify by subsitution that for any number C, y(x) = (x+C) cosx satisfies the given differential equation.
(b) Determine a value of the constant C so that y(3.14)= 0
Mostly this is just substitution:
a) $y(x) = (x + C)cos(x)$

Thus
$y^{\prime} = cos(x) - (x + C)sin(x)$

Thus the DEq says:
$(cos(x) - (x + C)sin(x)) + (x + C)cos(x) \cdot tan(x) = cos(x)$

$cos(x) - (x + C)sin(x) + (x + C)sin(x) = cos(x)$

$cos(x) = cos(x)$

So this is true.

b) We want a value for C such that $y(\pi) = 0$

$0 = (\pi + C)cos(\pi)$

$0 =-(\pi + C)$

$C = - \pi$

-Dan