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Euler’s method (solving 1st order differential equation)
Theory:- dy/dx=f(x,y) is 1st order equation, for approximation, a tangent at a given point will be fairly close to the original curve. Derivative at this point (x0,y0) is (dy/dx)x=x0 = f(x0,y0) Now, the tangent (y-y0)/(x-x0)=f(x0,y0) or, y=y0+f(x0,y0)*(x-x0)……..(I) if x1 is close enough to x0, we can say equation (I) can be the equation of the…
Newton-raphson method.
To find a root , f(x)=0 now, by Taylor series expansion and neglecting the higher orders. f(x)=f(x0)-f'(x0)*(x-x0)=0 or, x=x0-f(x0)/f'(x0) , x0 being the initial guess. generally:- xi=xi-1- f(xi)/f'(xi) i>=1 we iterate the values of x to reach as close to the actual root as possible.. Output:- so , generally Newton raphson method doesn’t require bounded…
Root finding by Bisection method.
Root of a function f(x) is that point where f(x) cuts x axis,(let’s say at x=c), which means f(c)=0. In question, we are told to find roots between two given points say a and b. A)) If f(a)*f(b)>0 , which is only possible if both f(a) and f(b) are positive, or both of them are…
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