- #1
Buffu
- 849
- 146
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##
Homework Equations
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 :(.