Statement (I) The elimination of arbitrary constants from ( , ) and ( ) from (y=( + + ) x ) results in a differential equation of order three. Statement (II) The elimination of arbitrary constants ( , ) and ( ) from (y= x+ x+ e^x ) results in a differential equation of order three.

Step-by-step solution

Statement I ( y=( + + ) x ) Differentiate both sides w.r.t. (x ), we get ( d y d x =( + + )=k ) Here, order (=1 ) So, statement ( I ) is false. Statements II : (y= x+ x+ e^x ) Differentiate both sides w.r.t. (x ), we get ( d y d x = + x+ e^x )...(i) Again, differentiating both sides w. r. t. (x ), we get ( aligned d^2 y d x^2 & =- x+ e^x (ii) d^3 y d x^2 & =- x+ e^x (iii) aligned ) Subtracting Eq. (ii) from Eq. (iii), we get ( aligned d^3 y d x^2 - d^2 y d x^2 & =- x+ e^x+ x- e^x & = x- x & = ( x- x) aligned ) Henc