Question

Find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h=0.01, then with step size h=0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2.

xy ′ = 12x − 3y,y(1) = 4,1 ≦ x ≦ 2

Answer 1

To find the exact solution of the initial value problem, we solve the differential equation and use Euler's method to approximate the solution. We start with the given initial condition, apply Euler's method for two different step sizes, and create a table comparing the approximate values with the actual solution.

Step-by-step explanation:

To find the exact solution of the given initial value problem, we can solve the differential equation xy' = 12x - 3y with the initial condition y(1) = 4. By rearranging the equation, we get y' = (12x - 3y) / x. This is a first-order linear ordinary differential equation.

Using the integrating factor method, we can rewrite the equation as dy/dx + (3/x)y = 12. The integrating factor is e^(∫3/x dx) = x^3. Multiplying both sides of the equation by x^3, we get x^3 dy/dx + 3x^2 y = 12x^3.

The left side of the equation is now the derivative of (x^3 y) with respect to x, so we can integrate both sides and solve for y. Integrating, we obtain x^3 y = 4x^4 - C, where C is the constant of integration. Rearranging, we have y = (4x - C/x^3). Plugging in the initial condition y(1) = 4, we can determine C = 0, and the particular solution is y = 4x - 4/x^3.

Now, to approximate the solution using Euler's method with step sizes h = 0.01 and h = 0.005, we initialize with x = 1 and y = 4. We then use the formula y_n+1 = y_n + h * f(x_n, y_n), where f(x_n, y_n) = (12x_n - 3y_n) / x_n. We iterate this calculation for each value of x in the given interval.