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

a$.$ To solve this, we should use the substitution

$u=$ help $($formulas$).$

With this substitution,

$y=$ help $(formulas).$

$y^{\text{'}}=$ help $(formulas)$

Enter derivatives using prime notation $($e$.$g$.,$ you would enter $y^{\text{'}}$ for $\frac{dy}{dx}).$

b$.$ After the substitution from the previous part, we obtain the following linear differential equation in $x,u,u^{\text{'}}.$

help $($equations$)$

c$.$ The solution to the original initial value problem is described by the following equation in $x,y.$

help $($equations$)$