Q. 2 - Calculus

Question

Create illustrations of the item(s) mentioned in the following. Look for examples that are distinct from those in the text.

(a) Two separable differential equations and two non-separable differential equations.

(b) A slope field that describes functions of the formula $f\left(x\right)=x^2+C$ as solution functions.

(c) Three scenarios from the real world that initial-value problems could be used to simulate.

Step-by-Step Solution

Answer

1. Examples of separable and non separable differential equations.
2. The function has a solution $C=-4,1,2.$
3. Three real-world scenarios using initial value problems. 

Part(a) Step 1: Given information

Differential equations are $\frac{dy}{dx}=4x^{2}y$, $\frac{dy}{dx}=e^{x-y}$,$\frac{dy}{dx}=2x+3y-4$, $\frac{dy}{dx}=\tan x+\ln y$

Part(a) Step 2: Explanation

When the variables in a differential equation can be split up and placed on either side of the equation, the equation is said to be separable. Separable differential equation examples include

i)$\frac{dy}{dx}=4x^{2}y$  ii)$\frac{dy}{dx}=e^{x-y}$ 

The variables in a non-separable differential equation cannot be divided and placed on either side of the equation. Non-separable differential equation examples include

i)$\frac{dy}{dx}=2x+3y-4$ii)$\frac{dy}{dx}=\tan x+\ln y$

Part(b) Step 1: Given information

The function $f\left(x\right)=x^2+C$

Part(b) Step 2: Explanation

Fracted $\frac{dy}{dx}=2xy-dx=2x$ is a differential equation whose solution functions of the form $f\left(x\right)=x^2+C$. Given here are the slope field and solution functions for three different values of $C (C=-4,1,2).$

Part(c) Step 1: Given information

The three real world situations

Part(c) Step 2: Explanation

The following three real-world scenarios could be modelled using initial value problems:

 i) A cold drink at $3^{0}C$ warms up to the room temperature of $30°\mathrm{C}$ using Newton's equation of heating and a proportionality constant of $k=0.05°C$ . The model is defined by the differential equation representing $\frac{dT}{dt}$ as a function of temperature $T.$

(ii) Assume a pond has $1000$ fish and that a logistic growth model is used to estimate their population $P\left(t\right)$, where the natural growth rate of fish is $k=0.1$ and the pond's carrying capacity is $10,000$ fish. A first-value problem is used to model the issue.

(iii) The quantity of an antibiotic diminishes at a rate proportional to the amount of antibiotic present in the body if $X\left(t\right)$ is the number of milligrams of an antibiotic present in the body $t$ hours after it is consumed. The initial-value problem is defined by a differential equation that includes the initial condition and the function $\frac{dX}{dt}$.