Question

Suppose that y1(t) and y2(t) are both solutions to the differential equation
dy/dt= a(t)y + b(t).
Write down a linear differential equation satisfied by y1(t) + y2(t).

Answer 1

\[y'_1=a(t)y_1+b(t)\]\[y'_2=a(t)y_2+b(t)\]\[\sum\quad\Rightarrow\quad y'_1+y'_2=a(t)y_1+a(t)y_2+2b(t)\]\[(y_1+y_2)'=a(t)(y_1+y_2)+2b(t)\quad,\quad z=y_1+y_2\]\[z'=a(t)z+2b(t)\]