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To solve the initial value problem (IVP) numerically: \[ \frac{dy}{dx} + y = g(t), \quad y(0) = 0 \] where \( g(t) \) is defined piecewise as: \[ g(t) = \begin{cases} 2, & 0 \leq t \leq 1 \\ 0, & t > 1 \end{cases} \] we can proceed with the following steps: ### 1. Solve the Differential Equation for \( 0 \leq t \leq 1 \): For \( 0 \leq t \leq 1 \), \( g(t) = 2 \). The differential equation becomes: \[ \frac{dy}{dt} + y = 2 \] This is a first-order linear ODE. The integrating factor \( \mu(t) \) is: \[ \mu(t) = e^{\int 1 \, dt} = e^t __________ _____ ______ ______ ___ _________ ___ _______ _________.
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