Solve $(t^{2}25)\frac{dx}{dt}=(x^{2}16),$ using separation of variables, given the inital condition $x(0)=4$

First, we must write it in the standard form $\frac{dx}{dt}=g(t)*h(x),$ which is:

$\frac{dx}{dt}=$ help $($formulas$).$

Now, separate the variables and integrate:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dtC$ help $($formulas$)$

After integrating, the equation becomes:

$$

$C$ help $($formulas$).$

Then solve for the general solution:

$x=$ help $($formulas$).$

Finally, use the initial condition to solve for the particular solution: $x_p=$ help $($formulas$).$