Differential Equation

1. Fundamental Definitions

Differential Equation, Order & Degree
Differential Equation: An equation which contains the derivative of a function.

Order: The order of a differential equation is the highest order of the derivative appearing in the equation.

Degree: The degree is the power of the highest ordered derivative present in the equation, provided the equation is expressed as a polynomial in derivatives (i.e., having a positive integer as the index of each derivative).

The order and degree of a differential equation are always positive integers.
Types of Solutions
General Solution: A solution in which the number of arbitrary constants is exactly equal to the order of the differential equation.

Particular Solution: A solution obtained from the general solution by giving particular values to the arbitrary constants.
Formation of Differential Equations
A differential equation can be formed from a given relation by eliminating the arbitrary constants.

This is achieved by differentiating the relation with respect to the independent variable as many times as the number of arbitrary constants present.

Family of circles above the X-axis and touching the origin

Separation of Variables
If the given differential equation can be written in the form:

$$ f(x)dx = g(y)dy $$

then the variables are separated. The general solution is obtained by integrating both sides:

$$ \int f(x)dx = \int g(y)dy + c $$
Homogeneous Differential Equations
A differential equation is homogeneous if all terms have the same degree.

To solve it, use the substitution $y = vx$ (or $x = vy$).

Differentiating $y = vx$ with respect to $x$ gives:

$$ \frac{dy}{dx} = v + x\frac{dv}{dx} $$
Linear Differential Equations
The standard form of a linear differential equation of the first order is:

$$ \frac{dy}{dx} + Py = Q $$

where $P$ and $Q$ are functions of $x$ or constants.

Step 1: Find the Integrating Factor (I.F.):

$$ \text{I.F.} = e^{\int P \,dx} $$

Step 2: The general solution is given by:

$$ y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.}) \,dx + c $$
Alternative Linear Form
For the linear differential equation of the form:

$$ \frac{dx}{dy} + px = Q $$

where $p$ and $Q$ are constants or functions of $y$, the Integrating Factor is:

$$ \text{I.F.} = e^{\int p \,dy} $$

The general solution is:

$$ x \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.}) \,dy + c $$

Population Growth and Growth of Bacteria
If the population $P$ increases at a rate proportional to the population present at that time, the differential equation is:

$$ \frac{dP}{dt} = kP $$

where $k > 0$ is the constant of proportionality representing growth.

Solving this gives the population at any time $t$:

$$ P = P_0 \cdot e^{kt} $$

(where $P_0$ is the initial population at $t=0$).
Radioactive Decay
The rate of disintegration is proportional to the amount $x$ present at that time:

$$ \frac{dx}{dt} = -kx $$

where $k > 0$ is the constant of proportionality, and the negative sign indicates that $x$ decreases as $t$ increases.

The solution giving the amount at any time $t$ is:

$$ x = x_0 e^{-kt} $$

where $x_0$ is the initial amount at $t=0$.

Half-Life Period: The time it takes for half the amount of the radioactive substance to disintegrate.
Newton's Law of Cooling
The rate of change of temperature of a cooling body at any time is proportional to the difference between the temperature of the body ($\theta$) and that of its surrounding medium ($\theta_0$).

$$ \frac{d\theta}{dt} = -k(\theta - \theta_0) $$

Solving this using the initial condition $\theta = \theta_1$ when $t = 0$ yields:

$$ \theta = \theta_0 + (\theta_1 - \theta_0)e^{-kt} $$

Inverted right circular cone (Surface Area & Volume Applications)