Question

Verify if the given equations are solutions to the differential equation.

dy/dt=k(y-a)

i) y=1-ae^(kt)
ii) y=a+Ce^(kt)

Answer 1

dy/dt=k(y-a) ... (1)

i) y=1-ae^(kt) (particular solution)
differentiate y w.r.t. t:
d(1-ae^(kt))/dt
= k(e^(kt)+a)-ka
= ky-ka
= k(y-a)
= right-hand side, QED

ii) y=a+Ce^(kt)
differentiate with respect to t:
dy/dt
=Cke^(kt)
=k(a+Ce^(kt))-ka
=ky-ka
=k(y-a)
=right-hand side QED