(Solved): Use the method for solving homogeneous equations to solve the following differential equation. \[ \ ...

Use the method for solving homogeneous equations to solve the following differential equation. \[ \frac{d x}{d t}=\frac{x^{2}+6 t \sqrt{t^{2}+x^{2}}}{t x} \] Express the equation in the form \( \frac{d x}{d t}=F\left(\frac{x}{t}\right) \). \[ F\left(\frac{x}{t}\right)= \] Use the method for solving Bernoulli equations to solve the following differential equation. \[ \frac{d y}{d x}+\frac{y}{x-12}=4(x-12) y^{\frac{1}{3}} \] Find an expression for \( v \) in the form \( v=y^{1-n} \) \[ v= \]