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## Description

** **

SESSION |
Nov-dec2023 |

PROGRAM |
BCA |

SEMESTER |
III |

course CODE & NAME |
DCA2101 & Computer Oriented Numerical Methods |

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**SET-I**

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**Show that**

**(a) **

**(b) **

**Ans 1.**

To prove the given identities, let’s start by defining the operators:

= Laplacian operator (also known as the second-order spatial derivative operator)

= Gradient operator (vector of first-order spatial derivatives)

Divergence operator (divergence of a vector field)

Scalar

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**b) To prove****:**

Let’s start with the left-hand side (LHS):

Using the definitions of the Laplacian and gradient operators, we have:

Expanding the terms, we can

Its Half solved only

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**Solve the system of equation by Gauss Elimination’s method**

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* *

*.*

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**Ans 2.**

Consider the given system of equations,

2x+y+4z=124

x+11y−z=338

x−3y+2z=20

Convert the system to matrix form,

The augmented matrix for the above matrix form,

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**Q3. Find the equation of the best fitting straight line for the data:**

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X |
1 |
3 |
4 |
6 |
8 |
9 |
11 |
14 |

Y |
1 |
2 |
4 |
4 |
5 |
7 |
8 |
9 |

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**Ans 3.**

To find the equation of the best-fitting straight line for the given data points, you can use linear regression. The equation of a straight line is typically represented as:

Y = mx + b

Where:

- Y is the dependent variable (in this case, the Y values).
- X is the independent variable (in this case, the X values).
- m is the

**Set-II**

**Q4. Evaluate f(15), given the following table of values:**

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x |
10 |
20 |
30 |
40 |
50 |

y = f(x) |
46 |
66 |
81 |
93 |
101 |

**Ans 4.**

To evaluate f(15) using the given table of values, you can use interpolation. Since the table provides values of y = f(x) for specific values of x, you can interpolate to find the value of f(15) which falls between x = 10 and x = 20.

We can use linear interpolation for this purpose. Linear interpolation assumes that the function f(x) varies linearly between two data points. Here’s how you can calculate f(15):

First, identify the two data

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**Q5. Use Taylor’s series method to solve the initial value problem:**

** for ** ** given that ** **.**

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**Ans 5.**

The Differential Equation

With initial condition

**Apply Runge-Kutta fourth order method to find an approximate value of y when x = 0.1 given that****,****.**

Ans 6.

To apply the Runge-Kutta fourth-order method to solve the given initial value problem, we need to follow these steps:

- Define the differential equation:
- Specify the initial