Solve the initial value problem y' + p(t)y = 0, y(0) = 1 where p(t) = {1, 0<= t <= 1 and 1, t > 1}

Question

This is problem 33 from section 2.4 of the book Elementary Differential Equations Ninth Edition by William E. Boyce and Richard C. DiPrima. I would really like just to have some of the steps to the solution if not all of them. Thank you.

Answer 1

Solve the initial value problem y' + p(t)y = 0, y(0) = 1 where p(t) = {1, 0<= t <= 1 and 1, t > 1}

We have the function p(t), defined by

p(t) = {1, 0<= t <= 1 and 1, t > 1}

which, in effect, is simply,

p(t) = 1, t >= 0.

The ODE then is,

y^'=-y,t≥0

Integrating both sides,

∫dy/y=-∫1 dt
ln(y)-ln(k)=-t
y/k=e^(-t)
y=k.e^(-t),t≥0

Initial condition

y(0)=1=k∙1=>k=1

Answer: y=e^(-t),  t≥0