Question

Solve the following differential equations.

Answer 1

(a) and (b) are separable. I've never seen the notation used in (c), could you explain what your text means by \({D_t}^{-1}(\cdots)\) ?

Answer 2

I am not sure what it means. Can you show me a or b? :)

Answer 3

\[\frac{dy}{dx}=-3ax+b~~\implies~~dy=(-3ax+b)\,dx\]
Integrate:
\[\int dy=\int(-3ax+b)\,dx~~\iff~~y=\cdots\]

Answer 4

According to this page ( http://webcache.googleusercontent.com/search?q=cache:J7H1oiWV9Y4J:www.codee.org/library/articles/linear-operators-and-the-general-solution-of-elementary-linear-ordinary-differential-equations/at_download/files%257Cfiles:000+&cd=9&hl=en&ct=clnk&gl=us ),
\[{D_t}^{-1}[f(t)]=\int f(t)\,dt\]
So for part (c), you're simply integrating:
\[{D_t}^{-1}[3\sqrt{t^7}]=\int 3\sqrt{t^7}\,dt\]