- #1

DivGradCurl

- 372

- 0

Please, help me find where I made a mistake. Here's what I've got:

[tex]S_0 = \$ 100,000 [/tex]

[tex]r = 0.09[/tex]

[tex]k(t) = \$ 800 \left( 1 + \frac{t}{120} \right) / \mbox{month}[/tex]

[tex]\frac{dS}{dt}=rS-k(t), \qquad S(0)=S_0[/tex]

[tex]\frac{dS}{dt}-rS=-k(t)[/tex]

[tex]\mu = \exp \left( -r \int dt \right) = e ^{-rt}[/tex]

[tex]S(t)=e^{rt}\int -800 \left( 1 + \frac{t}{120} \right) e ^{-rt} \: dt[/tex]

[tex]S(t)=e^{rt}\left( \frac{20e^{-rt}}{3r^2} + \frac{800e^{-rt}}{r} + \frac{20e^{-rt}t}{3r} + \mathrm{C} \right)[/tex]

[tex]S(t) = \frac{20t}{3r} + \frac{800}{r} + \frac{20}{3r^2} + \mathrm{C} e^{rt}[/tex]

[tex]t=0 \Rightarrow \frac{800}{r} + \frac{20}{3r^2} + \mathrm{C} = S_0 \Rightarrow \mathrm{C} = \frac{-20-2400r+3r^2 S_0}{3r^2} [/tex]

[tex]S(t)=\frac{20}{3r^2} - \frac{20e^{rt}}{3r^2} + \frac{800}{r} - \frac{800e^{rt}}{r} + S_0 e^{rt} + \frac{20t}{3r}[/tex]

[tex]S(t)=0 \Rightarrow t \approx - 131 \mbox{ months} [/tex]

which is WRONG!

Any help is highly appreciated.