InterviewSolution
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InterviewSolution
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Solve the following differential equation :`(("x"^2-1)"dy")/("dx")+2"x y"=2/(("x"^2-1))` |
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Answer» The given differential equation may be written as `(dy)/(dx) + { (2x)/(( x ^(2) - 1 ))} y = (2)/(( x ^(2) - 1 ) ^(2)) " " ` ... (i) This is of the form ` (dy)/(dx) + Py = Q, ` where `P = (2x)/((x ^(2) - 1 )) and Q = (2)/( ( x^(2) - 1 ) ^(2)) ` Thus, the given differential equatiion is linear. ` IF = e ^(int Pdx) = e ^(int (2x)/(( x ^(2)-1)) dx) = e ^( log (x^(2) -1) ) = ( x ^(2) - 1 )` So, the required solution is given by `yxx IF = int {Q xx IF } dx + C ` i.e., `y xx (x ^(2) -1 ) = int { (2)/(( x ^(2) - 1 ) ^(2)) xx ( x ^(2) - 1 ) } dx +C ` ` " " = 2 int (dx )/(( x ^(2) - 1 )) + C ` ` " " = 2 int (1)/(2) {(1)/(( x - 1 )) - (1)/(( x + 1 )) } dx + C ` [ by partial fraction ] ` " " = log |( x - 1 )/( x+ 1 )| + C .` Hence, ` y (x ^(2) - 1 ) = log |(x - 1 )/(x + 1)| + C` is the required solution. |
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