InterviewSolution
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InterviewSolution
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Let (dy)/(dx) + y = f(x) where y is a continuous function of x with y(0) = 1 and f(x) = {{:(e^(-x), if o le x le 2),(e^(-2),if x gt 2):} Which of the following hold(s) good ? |
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Answer» `y(1) = 2e^(-1)` `I.F. = e^(x)` `therefore"""Solution is ye"^(x) = INT e^(x) f(x) dx + C` now if `0 LE x le 2` then `ye^(x) = int e^(x) e^(-x) dx + C` `rArr""ye^(x) = x + C` `y(0) = 1, rArr C = 1` `therefore""ye^(x) = x + 1` `therefore""y=(x+1)/(e^(x))`, `therefore""y(1) = (2)/(e)` Also y' = `y'=(e^(x)-(x+1)e^(x))/(e^(2x))` `rArr""y'(1) = (e-2e)/(e^(2))=(-e)/(e^(2))=-(1)/(e)` If x `gt` 2 `ye^(x) = int e^(x-2) dx` `therefore""ye^(x) = e^(x-2) + C` `therefore""y = e^(-2) + Ce^(-x)` As y is continuous `therefore""underset(x rarr 2)("lim")(x+1)/(e^(x))=underset(x rarr 2)("lim")(e^(-2)+ Ce^(-x))` `therefore""3e^(-2) = e^(-2) + Ce^(-2) rArr C = 2` `therefore"""for x" gt 2` `y=e^(-2) + 2e^(-x)` `rArr""y(3) = 2e^(-3)+e^(-2) = e^(-2) (2e^(-1)+1)` `rArr""y'= -2e^(-x)` `rArr""y'(3) = - 2e^(-3)` |
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