- #1

karush

Gold Member

MHB

- 3,269

- 5

rewrite

$\quad\displaystyle y'-2y=-10$

obtain u(x)

$\quad\displaystyle u(t)=\exp\int -2 \, dt=e^{-2t}$

integrate using the boundaries:

$\quad\displaystyle \int_{y_0}^{e^{-2t}y}\,du=-10\int_{0}^{t}e^{-2v}\,dv$

resulting in

$\quad\displaystyle e^{-2t}y-y_0=-5\left(e^{-2t}-1\right)$

multiply thru by $e^{2t}$

$\quad\displaystyle y-y_0e^{2t}=5\left(1-e^{2t}\right)$

then

$\quad\displaystyle y=5\left(1-e^{2t}\right)+y_0e^{2t}$

then

$\quad\displaystyle y=5-5e^{2t}+y_0e^{2t}$

then

$\quad\displaystyle y=5-(y_0+5)e^{-2t}$

book answer

$\quad\displaystyle y=5+(y_0-5)e^{-2t}$ok seems to be a sign error someplace