Question

differential equations -> xe^-t dx/dt=t with the initial condition that x(0) = 1.

Answer 1

\[xe ^{-t} \frac{ dx }{ dt }=t\]

Answer 2

\[\int\limits x dx=\int\limits \frac{ t }{ e^{-t}} dt \]

Answer 3

is where im at now, wondering if this is correct

Answer 4

\[\large \int\limits\limits x dx=\int\limits\limits t e^t\; dt\]
Yup looks good so far.
By Parts on the right side :D

Answer 5

\[\frac{ x^2 }{ 2 }+c=e^t(t-1)\]

Answer 6

right o got that
now solving for c hold on

Answer 7

\[\frac{ 1 }{ 2 }+c=e^0(0-1)\]
c=\[-\frac{ 3 }{ 2 }\]

Answer 8

yay good job c: Pshhhh you didn't need any help with this one lol

Answer 9

so now i have
\[\frac{ x^2 }{ 2 }-\frac{ 3 }{ 2 }=e^t(t-1)\]

Answer 10

now the hard part for me is solving for x because the answer wants it in x=
soooo what can i do to the 3/2? would that be added to the -1? so the right side would be e^t(t+1/2)?

Answer 11

\[\frac{ x^2 }{ 2 }=e^t(t-1)+\frac{ 3 }{ 2 }\]

Answer 12

Multiply both sides by 2\[\large x^2-3=2e^t(t-1)\]Add 3 to both sides,\[\large x^2=2e^t(t-1)+3\]Square root,\[\large x=\pm \sqrt{2e^t(t-1)+3}\]

Answer 13

That's probably the route I would take. Get rid of the ugly fraction first :D

Answer 14

ohhhh my that is what i was going to ask next. but multiplying by two is probaly the rout my prof would have taken. Thank you @zepdrix !