Question

Estimate y(0.4) by Euler's method and R.K method when y'(x) = x +y2
. y(0)=0, and h=0.2.
(5 marks)​

Answer 1

Answer:

A number of problems in science and technology can be formulated

into differential equations. The analytical methods of solving differential

equations are applicable only to a limited class of equations. Quite often

differential equations appearing in physical problems do not belong to any

of these familiar types and one is obliged to resort to numerical methods.

These methods are of even greater importance when we realize that computing machines are now readily available which reduce numerical work

considerably.

Solution of a differential equation. The solution of an ordinary differential equation means finding an explicit expression for y in terms of a finite

number of elementary functions of x. Such a solution of a differential equation is known as the closed or finite form of solution. In the absence of such

a solution, we have recourse to numerical methods of solution.

Let us consider the first order differential equation

dy/dx  f(x, y), given y(x0

)  y0

(1)

to study the various numerical methods of solving such equations. In most

of these methods, we replace the differential equation by a difference equation and then solve it. These methods yield solutions either as a power series in x from which the values of y can be found by direct substitution, or

a set of values of x and y. The methods of Picard and Taylor series belong

to the former class of solutions. In these methods, y in (1) is approximated

by a truncated series, each term of which is a function of x. The information

about the curve at one point is utilized and the solution is not iterated. As

such, these are referred to as single-step methods.

The methods of Euler, Runge-Kutta, Milne, Adams-Bashforth, etc. belong to the latter class of solutions. In these methods, the next point on the

curve is evaluated in short steps ahead, by performing iterations until sufficient accuracy is achieved. As such, these methods are called step-by-step

methods.

Euler and Runga-Kutta methods are used for computing y over a limited range of x- values whereas Milne and Adams methods may be applied

for finding y over a wider range of x-values. Therefore Milne and Adams

methods require starting values which are found by Picard’s Taylor series

or Runge-Kutta methods

Step-by-step explanation:

Answer 2

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