Answer
(a) $y_{1} = ce^{at}$
(b) $y = ce^{at} + b/a$
(c) Agreement with (17), p.12.
Work Step by Step
5. Solve $dy/dt = ay-b$ by,
(a) Solving $dy/dt = ay$, first. Call this solution $y_{1}$.
Then
(b) Supposing that the original equation differs from $y_{1}$ by some constant, $k$, find a $k$ such that
(c) $y(t) = y_{1}(t) + k$ can be compared to the solution in the text in equation (17), p.12.
________
${Solution}$.
(a) $dy_{1}/dt = ay_{1}$
Formally, write this as follows:
$dy_{1} = a y_{1} dt$,
$\frac{dy_{1}}{y_1} = a dt$. So that
$\int\frac{dy_{1}}{y_{1}} = \int adt = a\int dt$.
So, $ \ln y_{1} = at + C$,
$y_{1} = e^{at+C} = e^{C}e^{at} = ce^{at}$.
(b) For
$y = y_{1} + k$,
guess that $k = b/a$. Then
$y = ca^{t} + b/a$
(c) Agreement with (17), p.12.
