Exercises

Solve the following differential equations:

1) \(\dot{y}+y=4\)
\(y\left(0\right)=0\)
\(\dot{y}+y=0\)
\(\dot{y}=-y\)
\(\frac{dy}{dt}=-y\)
\(dy=-ydt\)
\(-\frac{1}{y}dy=dt\)
\(-\int\frac{1}{y}dy=\int dt\)
\(-ln(y) +c_a = t + c_b\)                 With \(c_b - c_a = c\)
\(e^{ln(y)} = e^{-t} e^c\)                                    With \(A=e^{c}\)
\(y=Ae^{-t}\)
\(\dot{y}=-Ae^{-t}\)
Now we substitute using the conditions:
\(-Ae^{-0}+0=4\)
\(A=-4\)
\(y=4-\dot{y}\)
\(y=4+Ae^{-t}\)\(y=4+Ae^{-t}\)
\(y=4-4e^{-t}\)
2) \(\dot{y} =23\)
\(y(0)=1\)
\(\frac{dy}{dt} =23\)
\(dy=23dt\)
\(\int dy = 23\int dt\)