Solve the initial value problem $\displaystyle 2 y y^{\,\prime} + 3 = y^2 + 3 x$ with $y(0) = 7$.

1. To solve this, we should use the substitution
   $u =$ help (formulas)
   With this substitution,
   $y =$ help (formulas)
   $y^{\,\prime} =$ help (formulas)
   Enter derivatives using prime notation (e.g., you would enter $y^{\,\prime}$ for $\frac{dy}{dx}$).
   
   
2. After the substitution from the previous part, we obtain the following linear differential equation in $x, u, u^{\,\prime}$.
   help (equations)
   
   
3. The solution to the original initial value problem is described by the following equation in $x, y$.
   help (equations)