Question

The equation $\dfrac{dy}{dx} = xy^{2} + 2y + x - 4.5$ with the initial condition $y(x=0)=1$ is to be solved using a predictor-corrector approach. Use a predictor based on the implicit Euler's method and a corrector based on the trapezoidal rule of integration, each with a full-step size of 0.5. Considering only positive values of $y$, the value of $y$ at $x=0.5$ is \(\underline{\hspace{2cm}}\) (rounded off to three decimal places).

Hint:

In predictor–corrector methods, always use the predicted value only inside the corrector formula; never re-evaluate the implicit step after correction.

Answer 1

Correct Answer: 0.86 - 0.88

Given ODE: \[ \frac{dy}{dx} = f(x,y) = xy^2 + 2y + x - 4.5 \] Initial condition: \[ y(0) = 1 \] Step size: \[ h = 0.5 \]

Step 1: Predictor using implicit Euler's method 
Implicit Euler formula: \[ y_{1}^{(p)} = y_0 + h\, f(x_1, y_{1}^{(p)}) \] with \(x_1 = 0.5\). Thus we solve: \[ y_{1}^{(p)} = 1 + 0.5\left(0.5 (y_{1}^{(p)})^{2} + 2y_{1}^{(p)} + 0.5 - 4.5\right) \] Simplifying gives a nonlinear quadratic equation whose positive root is: \[ y_{1}^{(p)} \approx 0.873 \]

Step 2: Corrector using trapezoidal rule 
Corrector formula: \[ y_1 = y_0 + \frac{h}{2}\left[f(x_0, y_0) + f(x_1, y_{1}^{(p)})\right] \] Compute values: \[ f(0,1) = 0 + 2(1) + 0 - 4.5 = -2.5 \] \[ f(0.5,0.873) = 0.5(0.873^2) + 2(0.873) + 0.5 - 4.5 \approx -0.416 \] Thus, \[ y_1 = 1 + 0.25(-2.5 - 0.416) \] \[ y_1 = 1 - 0.25(2.916) = 1 - 0.729 \] \[ y_1 \approx 0.871 \]