Significant figures are used to establish the number which is presented in the form of digits. Significant Figures is an important topic of the arithmetic part of the mathematics/ quantitative aptitude section in any exam.
A linear equation or polynomial, with one or more terms, consisting of the derivatives of the dependent variable with respect to one or more independent variables is known as a linear differential equation. If you are appearing for an exam in which questions on calculus will be asked, then it becomes very important then you must know different methods of solving linear differential equations as it is a crucial part of differential calculus.
On that note, let’s discuss every concept of significant figures along with methods to solve any kind of linear differential equation to help you get edge in both arithmetics and differential calculus.
The significant figures of a given number are those significant or important digits, which convey the meaning according to its accuracy. For example, 6.658 has four significant digits. These substantial figures provide precision to the numbers.
Rules for Significant Figures
- Non-zero digits are all significant.
- All zeros between any two non-zero digits are important. For example, the number 108.0097 has seven significant digits.
- All zeros to the right of a decimal point, as well as all zeros to the left of a non-zero digit, are never significant. 0.00798, for example, has three significant digits.
- All zeros to the right of a decimal point are significant if they are not followed by a non-zero digit. 20.00, for example, has four significant digits.
- All of the zeros after the decimal point to the right of the final non-zero digit are significant. 0.0079800, for example, has five significant numbers.
Rounding Significant Figures
A number is rounded off to the required number of significant digits by leaving one or more digits to the right.
There are two rules for rounding off significant figures:
- To begin, we must determine up to which digit the rounding off should be conducted. If the number after the rounding off digit is less than 5, we must eliminate all of the numbers on the right side.
- However, if the digit next to the rounding off digit is more than 5, we must add 1 to the rounding off digit and disregard the remaining numbers on the right side.
Linear Differential Equations
A linear differential equation is a linear equation or polynomial with one or more terms consisting of the derivatives of the dependent variable with respect to one or more independent variables.
The expression for a general first-order differential equation is:
dy/dx + Py = Q where y is a function and dy/dx denotes a derivative
The linear differential equation’s solution yields the value of variable y.
dy/dx + 2y = sin x
dy/dx + y = ex
How to Solve Linear Differential Equation
Typically, two strategies are considered when solving a first-order linear differential equation.
Integrating Factor Method
In the conventional form of a linear differential equation:
y’ + a(x)y = 0
The formula then defines the integrating factor.
u(x) = exp (∫a(x)dx)
Multiplying the integrating factor u(x) on the left side of the equation yields the derivative of the product y(x)u (x).
The differential equation’s general solution is written as follows:
where C is a random constant
Method of Variation of a Constant
The integrating factor approach is comparable to this method. The first step is to find the general solution to the homogeneous equation.
y’ + a(x)y = 0
A constant of integration C is always present in the general solution of the homogeneous equation. We can substitute a certain unknown function C for the constant C. (x). We may determine the function C by substituting this solution into the non-homogeneous differential equation (x). The algorithm’s technique is known as the method of constant variation. However, both procedures get the same result.