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To solve these differential equations numerically, we would typically use methods like the Euler method, Runge-Kutta method, or finite difference methods, depending on the complexity of the equation. Given that the two equations are second-order ordinary differential equations (ODEs), we will describe the general approach. --- ### For (i) \(\left( \frac{dy}{dx} - 1 \right)^2 \left( \frac{d^2y}{dx^2} + 1 \right)^2 y = \sin^2\left(\frac{x}{2}\right) + e^x + x\): 1. Step 1: Rewriting the equation The given ODE is: \[ \left( \frac{dy}{dx} - 1 \right)^2 \left( \frac{d^2y}{dx^2} + 1 \right)^2 y = \sin^2\left(\frac{x}{2}\right) + e^x + x. \] This is a nonlinear second-order ______ __________ ___ ___ ______ ______ _________.
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