Introduction to Data Science Ordinary Differential Equations Introduction to Differential Equations

Let us recall the definition of ‘derivative’. A derivative, particularly in mathematics is the rate of change of function with respect to a variable. For example, in \(Figure  \space  1\), you will notice the rate of change of \(y\left(\triangle y\right)\) w.r.t variable \(x\) which shows the slope of curve or we can call it as \(f'(x)\). 

We denote derivative by \(\frac{dy}{dx}\) i.e., the change in \(y\) with respect to \(x\). If \(y(x)\)is a function, the derivative is represented as \(y'(x)\). Therefore, we can say that \(f'(x)=y'(x)= \frac{dy}{dx}\).

 

ODE1

\(Figure \space 1\)

Differential Equation

“In mathematics, a differential equation(DE) is an equation that relates one or more functions and their derivatives.In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.”

Let’s split the terms of Differential Equation. 

By now, we are aware that ‘Differentiate‘ means “Computing a derivative“, similarly ‘Differential Equation’ means “Equation containing a derivative“.

Few examples of DE:

  1. \( y’ = 2x + 3 \)
  2. \( y” + y’ + 10 = 3x \)

NOTE: 

  • For standard equations, the solution is a scalar value. (Example: \(11x=88\) which results to \(x=8\))
  • For Differential Equations, the solution is a function. (Example: \(\frac{\text{d}y}{\text{d}x}=y\) which results to \(y(x)=Ce^{x}\))

Order and Degree of Differential Equation

The ORDER of a differential equation is the order of the highest derivative or differential coefficient present in the equation.

Let us understand this by few examples:

  1. \( y’ + 5x + 6 = 0 \)     
  2. \( y” + 5y’ + 4x = 2 \)
In the first example, the order of DE is \(1\) because of single derivative \((y’)\) term. Similarly, for second DE the order is \(2\) because of the presence of double derivative \((y”)\) term.
 
Hence, we can classify DE on the basis of its Order as FIRST ORDER DIFFERENTIAL EQUATION and SECOND ORDER DIFFERENTIAL EQUATION.

 

The DEGREE of the differential equation is the power of highest order derivative in the equation.

  1. \((y”’)^{2}+(y”)^{3}+y’+9x = 24\)
  2. \(y”+(y’)^{2}+11x=7\) 
In the first example, the degree of the equation is  \(2\), whereas the order of the equation is  \(3\).
Similarly, the degree of the second equation is \(1\) and the order is \(2\).

Types of Differential Equation

There are three basic types of differential equations.

  1. Ordinary (ODEs) 
  2. Partial (PDEs)
  3. Differential-algebraic (DAEs) 

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