Given
dy/dt= -2y and y(2)=50
find
y(8)
Explanation
dy/dt= -2y
integrate both sides,
[tex]\begin{gathered} \int\frac{dy}{dt}=\int-2y \\ \\ \int\frac{dy}{y}=\int-2dt \\ \\ \ln|y|=-2t+c..................(1) \\ \end{gathered}[/tex]given , y(2) = 50
so ,
[tex]\begin{gathered} \ln|50|=-2(2)+c \\ \ln|50|+4=c \end{gathered}[/tex]now substitute the value of this see in 1 equation ,
[tex]\begin{gathered} \ln|y|=-2t+\ln|50|+4 \\ \\ \frac{\ln|y|}{\ln|50|}=-2t+4 \\ \\ \ln|y|-\ln|50|=-2t+4 \\ \\ \ln\frac{y}{50}=-2t+4 \\ \\ \frac{y}{50}=e^{-2t+4} \\ \\ y=50e^{-2t+4} \end{gathered}[/tex]now , put t = 8 , to find the value of y(8) ,
[tex]\begin{gathered} y(8)=50e^{-2(8)+4} \\ \\ y(8)=50e^{-16+4} \\ y(8)=50e^{-12} \\ y(8)=0.00030721061\approx0.00031 \end{gathered}[/tex]Final Answer
Hence , the value of y(8) is 0.00031
