Introduction to Differential Equations.
Solve the following differential Equations:
1) \(\frac{dy}{dt}+y\left(t\right)\ =\ 4\ \ ;\ \ y\left(0\right)=0\)
Homogeneous
\(\frac{dy}{dt}+y\left(t\right)\ =\ 0\)
\(\frac{dy}{dt}=\ -y\left(t\right)\)
\(dy\ =\ -y\left(t\right)dt\)
\(\int dy=\int -y\left(t\right)dt\)-
\(y\left(t\right)=e^{-t+c}\)
\(y(t)=e^{-t}e^c\)
\(y(t) = Ae^{-t}\)
Particular
\(0+y(t) = 4\)
\(y\left(t\right)=4\)
General Solution
\(y\left(t\right)=Ae^{-t}+4\)
Specific Solution
\(y\left(0\right)=Ae^{-t}+4=0\)
\(y\left(0\right)=Ae^{-\left(0\right)}=-4\)
\(y\left(0\right)=A\left(1\right)=-4\)
\(A=-4\)
\(y\left(t\right)=-4e^{-t}+4\)
2) \(\frac{dy}{dt}=\ 23\ \ ;\ \ y\left(0\right)=1\)
\(dy\ =\ 23dt\)
\(\int dy\ =\ \int 23dt\)
\(y\left(t\right)\ =\ 23t+c\)
specific solution