Find y"(0) and y'''(0)?

Question

Let y(t) be a function which satisfies the differential equation y"(t)+(1+t)y'(t)+(1+t+t^2)y(t)=0, y(0)=1, y'(0)=0. Find y"(0) and y'''(0).

Answer 1

Plug in t = 0,

y''(0) + y'(0) + y(0) = y''(0) + 0 + 1 = 0 => y''(0) = -1

Differentiate both sides of y"(t)+(1+t)y'(t)+(1+t+t^2)y(t)=0 with respect to t,

y'''(t) + y'(t) + (1+t)y''(t) + (1+2t)y(t) + (1+t+t^2)y'(t) = 0

Plug in t = 0,

y'''(0) + y'(0) + y''(0) + y(0) + y'(0) = y'''(0) + 0 - 1 + 1 + 0 = 0

y'''(0) = 0