Solve the given Initial value problem.

Question

(e^8y-y)cos6t(dy/dt)=e^y sin6t, y(0)=0

Let y(t) be the solution of the initial value problem. If y(t₀)=1, find the value of ln |cos6t₀|

With the answer rounded to exactly 4 digits after the decimal point.

Answer 1

∫(e^7y - ye^-y)dy = ∫tan6tdt; e^7y/7 + ye^-y + e^-y = - ln(cos6t)/6 + C; 1/7 + 1 = C; C = 8/7;

e^7y/7 + ye^-y + e^-y = - ln(cos6t)/6 + 8/7. Let t = t_0. Then e⁷/7 + 2/e = - ln(cos6t₀ + 8/7;

ln(cos6t₀) = - (e⁷/7 + 2/e - 8/7) = - 156.2548