In my textbook, the authors tried to solve the differential equation: $dy/dt+ay=g(t)$, where $a$ is a constant and $g(t)$ is a function. Why the authors of my textbook tend to leave the answer in terms of definite integral instead of indefinite integral if the indefinite integral cannot be evaluated in terms of elementary functions (the authors leave the answer as $y=e^{-at}\int_{t_0}^t e^{as} g(s) ds + ce^{-at}$ (by using s as a dummy variable of integration, c as an arbitrary constant, $t_0$ as a convenient lower limit of integration) instead of $y=e^{-at}\int e^{at} g(t) dt + ce^{-at}$ if $\int e^{at} g(t) dt$ cannot be evaluated in terms of elementary functions). Please refer to the text:
On pages: 5-6 (34-35) (the numbers in the parentheses are the actual page numbers), the authors wrote "... For many simple functions g(t), we can evaluate the integral in Eq. (23) and express the solution y in terms of elementary functions, as in Example 2. However, for more complicated functions g(t), it is necessary to leave the solution in integral form. In this case $y=e^{-at}\int_{t_0}^t e^{as} g(s) ds +ce^{-at}$..."
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