I found it at the end of this website Differential Equations - Laplace Transforms
I felt good about the other practice problems on the site but this one is over my head. Maybe it can be written with more in depth steps? Thanks.
I found it at the end of this website Differential Equations - Laplace Transforms
I felt good about the other practice problems on the site but this one is over my head. Maybe it can be written with more in depth steps? Thanks.
Attachment 38202
Sorry.. Dont know what going on with these multiple uploads of same picture. So is this reasoning sound? specifically steps 2 to 3 because I think that was what was most confusing. Also, the equation says that you need f(t) but we are using f'(t). Is this still ok?
I don't know how you got all three of them there but if you need to remove some then scroll down the page for a bit until you see the button "manage attachments." Click on the image you want removed and you'll see a little x in the top right. Click that and the image is removed. Remember to save the post afterward.
-Dan
The Laplace transform of f(x) is defined as $L(f)= \int_0^\infty f(t)e^{-st} dt$. To find the Laplace transform of f'(t), we can use "integration by parts". $L(f')= \int_0^\infty f'(t)e^{st}dt$. Use integration by parts with $u(t)= e^{-st}$ and $dv(t)= f'(t)dt$. Then $du= -se^{-st}$ and $v= f(t)$ so $\int_0^\infty f'(t)e^{-st}dt= \left[e^{-st}f(t)\right]_0^\infty+ s\int_0^\infty f(t)e^{-st}dt= f(0)+ sL(f)$.