1.) The independent variable x is missing in the given differential equation. Proceed as in Example 2 and solve the equation by using the substitution u = y'.(y + 7)y'' = (y' )22.) The independent variable x is missing in the given differential equation. Proceed as in Example 2 and solve the equation by using the substitution u = y'.y'' + 6y(y')3 = 0

The solution to the differential equation y'(y + 7)y'' = (y')²

y = Ae^(Kx) - 7

Step-by-step explanation:

Given the differential equation

y'(y + 7)y'' = (y')² (1)

We want to solve using the substitution u = y'.

Let u = y'

The u' = y''

Using these, (1) becomes

u(y + 7)u' = u²

u' = u²/u(y + 7)

u' = u/(y + 7)

But u' = du/dy

So

du/dy = u/(y + 7)

Separating the variables, we have

du/u = dy/(y + 7)

Integrating both sides, we have

ln|u| = ln|y + 7| + ln|C|

u = e^(ln|y + 7| + ln|C|)

= K(y + 7)

But u = y' = dy/dx

dy/dx = K(y + 7)

Separating the variables, we have

dy/(y + 7) = Kdx

Integrating both sides

ln|y + 7| = Kx + C1

y + 7 = e^(Kx + C1) = Ae^(Kx)

y = Ae^(Kx) - 7