Question

Question: Reread this section and classify each mathematical model as linear or nonlinear.

Short answer

The linear models are $\frac{\text{dP}}{\text{dt}}\text{= kP,}\frac{\text{dA}}{\text{dt}}\text{= kA,}\frac{\text{dT}}{\text{dt}}\text{= k}\left({\text{T -T}}_{\text{m}}\right)\text{,}\frac{\text{dA}}{\text{dt}}\text{+}\frac{\text{1}}{\text{100}}\text{A = 6,}$$\text{L}\frac{{\text{d}}^{\text{2}}\text{q}}{{\text{dt}}^{\text{2}}}\text{+ R}\frac{\text{dq}}{\text{dt}}\text{+}\frac{\text{1}}{\text{C}}\text{q = E(t),}$$\frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{= - g,m}\frac{\text{dv}}{\text{dt}}\text{= mg - kv,}$and$\text{m}\frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{+ k}\frac{\text{ds}}{\text{dt}}\text{= mg}$

The nonlinear models are$\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{kx}(\mathrm{n}+1-\mathrm{x}),\frac{\mathrm{dX}}{\mathrm{dt}}=\mathrm{k}(\mathrm{\alpha }-\mathrm{X})(\mathrm{\beta }-\mathrm{X}),$and$\frac{\mathrm{dh}}{\mathrm{dt}}=\frac{{\mathrm{A}}_{\mathrm{h}}}{{\mathrm{A}}_{\mathrm{w}}}\sqrt{2\mathrm{gh}}$

Step-by-step solution

Define a derivative of the function.

Classification by linearity: An nth-order ordinary differential equation is linear when it has the following form
${\text{a}}_{\text{n}}\text{(x)}\frac{{\text{d}}^{\text{n}}\text{y}}{{\text{dx}}^{\text{n}}}{\text{+a}}_{\text{n - 1}}\text{(x)}\frac{{\text{d}}^{\text{n - 1}}\text{y}}{{\text{dx}}^{\text{n - 1}}}{\text{+ L +a}}_{\text{1}}\text{(x)}\frac{\text{dy}}{\text{dx}}{\text{+a}}_{\text{0}}\text{(x)y = g(x)}$

Determine the linear models

Based on the above definition, there are eleven linear different mathematical models. They are,

Determine the nonlinear models

Based on the above definition, there are three nonlinear different mathematical models. They are,