Sometimes a change of variable can be used to convert a differential equation $y^{\,\prime} = f(t,y)$ into a separable equation. One common change of variable technique is as follows.

1. Consider a differential equation of the form $y^{\,\prime} = f(\alpha t + \beta y + \gamma)$, where $\alpha, \beta$, and $\gamma$ are constants. Use the change of variable $z = \alpha t + \beta y + \gamma$ to rewrite the differential equation as a separable equation of the form $z^{\,\prime} = g(z)$.
   
   

Solve the initial value problem

$$y^{\,\prime} = (t+y)^{2} - 1, \ \ \ y(1) = 5.$$

(a) $g(z) =$ help (formulas)

(b) $y(t) =$ help (formulas)