Differential Equations and Modelling

Integrate directly. The constant of integration is required for a general solution.

Separating gives $\ln|y| = 3t + C$, so $y = Ae^{3t}$. This is the standard growth equation.

$\text{IF} = e^{\int \frac1x dx} = e^{\ln x} = x$.

Factorising: $(m-2)(m+5) = 0$. The general solution is $y = Ae^{2x} + Be^{-5x}$.

$m^2 + 9 = 0$ gives $m = \pm 3i$, so $\alpha = 0$ and $\beta = 3$. Purely imaginary roots give an undamped oscillation.

Separating gives $\ln|\theta - 20| = -kt + C$, so $\theta - 20 = Ae^{-kt}$. The temperature approaches $20°$ without ever reaching it.