Solve the initial value problem $y{y}^{\text{'}}+x=\sqrt[\phantom{2}]{x^2+y^2}$ with $y(2)=\sqrt[\phantom{2}]{32}.$

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

$u=$ help $(formulas)$

$u^{\text{'}}=$ 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 nart, 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$)$