Question

Show that Y=ln|cos3/cosx|+5 is the solution to the initial value problem dy/dx=tanx, f(3)=5

Answer 1

dy=(sinx/cosx)dx.

Let u=cosx, then du=-sinxdx so dy=-du/u, and integrating: y=-ln(au) where a is a constant.

Therefore y=ln|1/acosx|=ln|secx/a|.

When x=3, y=5, 5=ln|sec(3)|+c, where c=ln(1/a).

So c=5-ln|sec(3)|=5+ln|cos(3)| and y=ln|cos(3)/cosx|+5 QED.

Comments

Thank you.