Question

Percentage Error Let \(y=f(x)\) be solution to the initial value problem \(d y / d x=2 x-1\) such that \(f(2)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=-0.1\) is used to ap- proximate \(f(1.6) .\)

Short answer

Percentage error is calculated using the values obtained from the exact equation and the Euler's method, then using these to find the difference between the exact and approximate values relative to the exact value, multiplied by 100 to put it in terms of percentage. Using mathematical calculations, we can solve this value numerically.

Step-by-step solution

Solve The Differential Equation

Given the differential equation \(dy/dx = 2x - 1\) with the initial condition \(f(2)=3\), we can find its general solution by integrating both sides. After Integrating, the solution to the equation becomes \(f(x) = x^2 - x + c\). We can substitute \(x = 2\) and \(f(2) = 3\) into the equation to find \(c = -1\). Thus the exact solution for the differential equation is \(f(x) = x^2 - x - 1\).

Euler's Method Implementation

Euler's method is a numerical method used to approximate solutions of first order differential equations. It is defined by the recurrence relation \(y_{n+1} = y_n +hf(x_n, y_n)\), where in our case, \(h=-0.1\) and \(f(x, y) = 2x- 1\). Starting with \(x_0=2\), \(y_0=f(2)=3\) we apply the recurrence relation five times to reach \(x=1.6\).

Calculate The Percentage Error

The percentage error is calculated using the formula \((\text{exact value} - \text{approximate value}) / \text{exact value} * 100\%\). In this case, the exact value is given by \(f(1.6) = (1.6)^2 - 1.6 -1\) and the approximate value is given by the result at \(x=1.6\) from Euler's method.