Question

2: If y1=1.7,h=0.1,f(x1, y1)-1.2 then by Euler's method
the alue of y2 is
3417275302​

Answer 1

Answer:

Ordinary Differential Equations Euler’s Method: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

3. Ordinary Differential Equations Euler’s Method: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 The Taylor’s series is y(x) = y(x0) + (x − x0) 1! y (x0) + (x − x0)2 2! y (x0) + . . . - - - (1) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

4. Ordinary Differential Equations Euler’s Method: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 The Taylor’s series is y(x) = y(x0) + (x − x0) 1! y (x0) + (x − x0)2 2! y (x0) + . . . - - - (1) Now substituting h = x1 − x0 in eq (1), we get Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

5. Ordinary Differential Equations Euler’s Method: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 The Taylor’s series is y(x) = y(x0) + (x − x0) 1! y (x0) + (x − x0)2 2! y (x0) + . . . - - - (1) Now substituting h = x1 − x0 in eq (1), we get y(x1) = y(x0) + hy (x0) + h2 2! y (x0) + . . . Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

6. Euler’s Method If h is chosen small enough then we may neglect the second and higher order term of h. Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

7. Euler’s Method If h is chosen small enough then we may neglect the second and higher order term of h. y1 = y0 + hf(x0, y0) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

8. Euler’s Method If h is chosen small enough then we may neglect the second and higher order term of h. y1 = y0 + hf(x0, y0) Which is Euler’s first approximation. Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

9. Euler’s Method If h is chosen small enough then we may neglect the second and higher order term of h. y1 = y0 + hf(x0, y0) Which is Euler’s first approximation. The general step for Euler method is Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

10. Euler’s Method If h is chosen small enough then we may neglect the second and higher order term of h. y1 = y0 + hf(x0, y0) Which is Euler’s first approximation. The general step for Euler method is yi+1 = yi + hf(xi, yi) where i = 0, 1, 2.... Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

11. Euler’s Method Ex.: Use Euler’s method to find y(1.6) given that dy dx = xy 1 2 , y(1) = 1 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations