Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use
h = 0.1 and then use h = 0.05.
y’ = (x – y)2, y(0) = 0.7; y(0.5)
y(0.5) ˜
(h = 0.1)
y(0.5) ˜
(h = 0.05)
The Correct Answer and Explanation is:
Using Euler’s method to approximate the solution of the differential equation y′=(x−y)2,y(0)=0.7y’ = (x – y)^2, \quad y(0) = 0.7
at x=0.5x = 0.5, we compute the following:
- With step size h=0.1h = 0.1, y(0.5)≈0.8743y(0.5) \approx \mathbf{0.8743}
- With step size h=0.05h = 0.05, y(0.5)≈0.8632y(0.5) \approx \mathbf{0.8632}
Explanation
Euler’s method is a numerical technique used to approximate solutions to first-order ordinary differential equations (ODEs). It is especially useful when exact solutions are difficult or impossible to find. The method works by stepping through the independent variable (here, xx) in small increments (called the step size, hh), using the derivative function to estimate the next value of the dependent variable (here, yy).
The equation we are solving, y′=(x−y)2y’ = (x – y)^2, is nonlinear and does not have a straightforward closed-form solution. Given the initial condition y(0)=0.7y(0) = 0.7, we apply Euler’s method over the interval from 0 to 0.5.
The update rule for Euler’s method is: yn+1=yn+h⋅f(xn,yn)y_{n+1} = y_n + h \cdot f(x_n, y_n)
where f(x,y)=(x−y)2f(x, y) = (x – y)^2. We repeat this step until we reach the desired xx-value, in this case 0.5.
Using h=0.1h = 0.1, we take 5 steps. This gives a rough approximation, yielding y(0.5)≈0.8743y(0.5) \approx 0.8743. When we halve the step size to h=0.05h = 0.05, we take 10 smaller steps. This increases accuracy, resulting in a slightly lower value, y(0.5)≈0.8632y(0.5) \approx 0.8632.
This behavior is typical: smaller step sizes reduce truncation error, leading to more accurate approximations. However, this also increases computational effort. Comparing the two results shows the convergence of the approximation and illustrates the tradeoff between accuracy and efficiency in numerical methods.
