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**1. Homework Statement**

##\dfrac{dy}{dx} + y = f(x)##

##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}##

##y(0) = 0##

**2. Homework Equations**

**3. The Attempt at a Solution**

Integrating factor is ##e^x##

##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)##

##\displaystyle ye^x = \int e^x f(x) dx = \begin{cases} \int 2e^x dx = 2e^x + C \qquad x \in [0, 1) \\ \int 0 dx = C^\prime \qquad x \ge 1 \end{cases}##

Solving for inital value,

##ye^x = \begin{cases} 2(e^x - 1) \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}##

The answer is ##ye^x = \begin{cases} 2(e^x - 1) \qquad x \in [0, 1) \\ 2(e - 1) \qquad x \ge 1 \end{cases}##

, which is way off.

I think the fault is in my answer this time but I can't point it :(.