Using Taylor’s series method, compute y(0.2) to three places of deci-mals from 1 2dy xydx   given that y(0) 0.

To compute y(0.2) using Taylor's series method, we need to follow these steps:

Step 1: Determine the differential equation and initial condition: The given differential equation is 12dy/dx - xy = 0, and the initial condition is y(0) = 0.

Step 2: Express the differential equation in terms of y and its derivatives: We can rewrite the equation as dy/dx = (xy)/12.

Step 3: Find the derivatives of y with respect to x: We need to find the first few derivatives of y with respect to x. Let's denote the nth derivative of y as y^(n).

The first derivative is dy/dx = (xy)/12. The second derivative is d^2y/dx^2 = (d/dx)(dy/dx) = (d/dx)((xy)/12) = (y/12) + (x/12)(dy/dx). The third derivative is d^3y/dx^3 = (d/dx)(d^2y/dx^2) = (d/dx)((y/12) + (x/12)(dy/dx)) = (1/12)(dy/dx) + (x/12)(d^2y/dx^2).

Step 4: Evaluate the derivatives at x = 0 and y = 0: Since y(0) = 0, all the derivatives of y with respect to x evaluated at x = 0 will be 0.

Step 5: Write the Taylor series expansion: The Taylor series expansion of y(x) around x = 0 is given by: y(x) = y(0) + (dy/dx)(0)x + (d^2y/dx^2)(0)(x^2/2!) + (d^3y/dx^3)(0)(x^3/3!) + ...

Since all the derivatives evaluated at x = 0 are 0, the Taylor series simplifies to: y(x) = 0 + 0 + 0 + ...

Step 6: Approximate y(0.2) using the Taylor series: Since all the terms in the Taylor series expansion are 0, y(0.2) will also be 0.

Therefore, y(0.2) is approximately 0.