Solve

$\frac{dy}{dx}=ytanx$

using separation of variables given the inital condition $y(0)=11.$ Assume the function is defined for $\frac{dy}{dx}=g(x)*h(y)\frac{dy}{dx}={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dy={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dxC$ help $(formulas)=,C$ help $(formulas)Cy=,$ help $(formulas)y_p=,$ help $(formulas)-\frac{}{2}.$

First, $we$ must write $itin$ the standard form $\frac{dy}{dx}=g(x)*h(y),$ which $is$:

$\frac{dy}{dx}=$

$$

$$ help $(formulas).$

Now, separate the variables and integrate:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dy={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dxC$ help $(formulas)$

After integrating, the equation becomes:

$=,C$ help $(formulas)$

Then solve for the general solution, using $Cas$ the aribtary constant:

$y=,$ help $(formulas)$

Finally, use the initial condition $to$ solve for the particular solution:

$y_p=,$ help $(formulas)$