Question

The dependent variable y is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution u = y'.
y'' + (y' )^2 + 4 = 0

Answer 1

Answer:

y = In | cos(2x + c ) | + c

Step-by-step explanation:

y" + (y')^2 + 4 = 0

substituting u = y'

u' + u^2 + 4 = 0

hence : u' = - (u^2 + 4 )

       $\frac{u'}{-(u^2 + 4)}$ = 1 ------- (1)

integrating both sides of the equation 1

$1/2 \int\limits^1_1 {\frac{2du}{(u^2+4)} } \, = x + c$

x + c = $- \frac{1}{2} arc tan (\frac{u}{2} )$   hence  u = -2 tan(2x + c )

remember u = y'

y' = -2 tan(2x + c) ------ (2)

integrating both sides of the equation 2

y = ∫ $\frac{-sin u}{cos u } du$

therefore Y = In | cosu | + c

 y = In | cos(2x + c ) | + c