Answer
$y_0=21/8+3e^{4/3}/4\approx 5.470$
Work Step by Step
Finding the integrating factor:
$\mu(t)=\exp(\int 2/3\ dt)$
$\mu(t)=e^{2t/3}$
Finding the y-function:
$y=\frac{1}{\mu(t)}[\int \mu(s)(1-s/2)ds+c]$
$y=e^{-2t/3}[-3/8\ e^{2t/3}(2t-7)+c]$
$y=-3/8\ (2t-7)+ce^{-2t/3}$
$y(0)=21/8+c=y_0\rightarrow c=y_0-21/8$
If y touches but not crosses the t-axis, 0 is a local minimum so:
$y'(t_m)=0,\ y(t_m)=0\rightarrow 0+2/3\cdot0=1-t_m/2\rightarrow t_m=2$
$y'(0)=-3/8(2)+(y_0-21/8)e^{-2\cdot 2/3}=0$
$-3/4+(y_0-21/8)e^{-4/3}=0$
$y_0=e^{4/3}(21/8e^{-4/3}+3/4)$
$y_0=21/8+3e^{4/3}/4\approx 5.470$
