Math Solved Question(3rd Semester)

1. Find Third Dividend Difference with  arguments 2, 4, 9, 10 of the function f(x) = x³-2x.

Solution:-

The given function is f(x) = x³-2x

Given arguments - 2,4,9,10

Let x₀ = 2, x₁ = 4, x₂ = 9, x₃ = 10

∴ f(x₀) = f(2) = 2³ - 2×2 = 4

∴ f(x₁) = f(4) = 4³ - 2×4 = 56

∴ f(x₂) = f(9) = 9³ - 2×9 = 711

∴ f(x₃) = f(10) = 10³ - 2×10 = 980

∴ First Dividend Difference -

i) Δₓ₁f(x₀) = f(x₀,x₁) = {f(x₁) - f(x₀)}/(x₁ - x₀) = (56-4)/(4-2) = 26

ii) Δₓ₂f(x₁) = f(x₁,x₂) = {f(x₂) - f(x₁)}/(x₂ - x₁) = (711-56)/(9-4) = 131

iii) Δₓ₃f(x₂) = f(x₂,x₃) = {f(x₃) - f(x₂)}/(x₃ - x₂) = (980-711)/(10-9) = 269

∴ Second Dividend Difference -

i) Δ₍ₓ₁,ₓ₂₎f(x₀) = f(x₀,x₁,x₂) = {f(x₁,x₂) - f(x₀,x₁)}/(x₂ - x₀) = (131-26)/(9-2) = 15

ii) Δ₍ₓ₂,ₓ₃₎f(x₁) = f(x₁,x₂,x₃) = {f(x₂,x₃) - f(x₁,x₂)}/(x₃ - x₁) = (269-131)/(10-4) = 23

∴ Third Dividend Difference -

Δ₍ₓ₁,ₓ₂,ₓ₃₎f(x₀) = f(x₀,x₁,x₂,x₃) = {f(x₁,x₂,x₃) - f(x₀,x₁,x₂)}/(x₃ - x₀)

= (23-15)/(10-2) = 1.Answer.

Example: Use Euler’s procedure to  find y(0.4) from the differential equation

Sol:

Given equation  dy/dx = xy

Here 

 We  break  the  interval in four steps.

 So that 

By Euler’s formula

For n=0 in equation (i) we get, the first approximation

n=1 in equation (i) we obtain

Put=2 in equation (i) we  get, the third approximation

Put n=3 in equation (i) we  get, the fourth approximation

Hence y(0.4)  =1.061106.