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Let g(x) = sin x for 0 <= x < pi/2 and g(x) = cos x for x >= pi/2. A continuous function y = f(x) satisfies 5*dy/dx + 5*y = g(x) with f(0) = 0. Which statement is correct?
- f(pi/4) = e^(-pi/4)/10
- f(pi/4) = (e^(-pi/4) - 1)/10
- f(pi/2) = (e^(-pi/2) + 1)/10
- f(pi/2) = e^(-pi/2)
Correct answer: f(pi/4) = (e^(-pi/4) - 1)/10
Solution
Solving the linear ODE on the first branch with f(0)=0 gives f(pi/4) = (e^(-pi/4) - 1)/10.
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