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) $\displaystyle y(x) = (x + C)cos(x)$

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

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

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

$\displaystyle cos(x) = cos(x)$

So this is true.

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

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

$\displaystyle 0 =-(\pi + C)$

$\displaystyle C = - \pi$

-Dan