InterviewSolution
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InterviewSolution
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If `y(x)` is solution of `x(dy)/(dx)+2y=x^(2), y(1)=1` then value of `y(1/2)=` (a) `-(49)/(16)` (b) `(49)/(16)` (c) `(45)/(8)` (d) `-(45)/(8)`A. (a) 13/16B. (b) 1/4C. (c) 49/16D. (d) 7/64 |
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Answer» Correct Answer - (c) Given differential equation can be rewritten as `dy/dx + (2/x)cdoty=x,` which is a linear differential equation of the form `dy/dx + Py =Q.` where`P=2/x and Q=x` Now, integrating factor `(IF) = e^(int 2/x dx) =e^(2log x) =e^(log x^(2)=x^(2)` `[therefore e^(logf(x))=f(x)]` and the solution is given by `y(IF)= int (QxxIF) dx + C` ` rArr yx^(2)=int x^(3) dx +C` ` rArr yx^(2)= x^(4)/4+C …(i)` Since, it is given that Y = 1 when x = 1 `therefore` From Eq. (i). we get `1=1/4+c rArr c =3/4 …(ii)` `therefore 4x^(2)y=x^(4)+3` [using Eqs. (i) and (ii) ] `rArr y=(x^(4)+3)/(4x^(2))` Now,`y(1/2)=(1/16+3)/(4xx1/4)= 49/16` |
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