UNIT I

Differential Equations
_________________________________________________________________________
𝑑2𝑦
((Q))1_// The solution of D.E. 𝑑𝑥 2 + 9𝑦 = 0 is
((A)) 𝑐1 cos2𝑥 + 𝑐2 sin2𝑥
((B)) (𝑐1 𝑥 + 𝑐2 )𝑒 −3𝑥
((C)) 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 −3𝑥
((D)) 𝑐1 cos3𝑥 + 𝑐2 sin3𝑥
((E))D
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E. 𝑑𝑥 2 − 𝑑𝑥 − 2𝑦 = 0 is
((A)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 𝑥
((B)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 −𝑥
((C)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 𝑥
((D)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 −𝑥
((E))B
((F))

((Q))1_// If 𝑚1 = 𝛼 + 𝑖𝛽 ((A))nd 𝑚2 = 𝛼 − 𝑖𝛽 are two complex roots of auxiliary equation of second order
DE 𝜙(𝐷)𝑦 = 0 then it’s solution is
((A)) 𝑒 𝛽𝑥 [𝑐1 cos𝛼𝑥 + 𝑐2 sin𝛼𝑥]
((B)) 𝑒 𝛼𝑥 [(𝑐1 𝑥 + 𝑐2 )cos𝛽𝑥 + (𝑐3 𝑥 + 𝑐4 )sin𝛽𝑥]
((C)) 𝑐1 𝑒 𝛼𝑥 + 𝑐2 𝑒 𝛽𝑥
((D)) 𝑒 𝛼𝑥 [𝑐1 cos𝛽𝑥 + 𝑐2 sin𝛽𝑥]
((E))D
((F))

((Q))1_// If the complex roots 𝑚1 = 𝛼 + 𝑖𝛽 and 𝑚2 = 𝛼 − 𝑖𝛽 of auxillary equation of fourth order DE

𝜙(𝐷)𝑦 = 0 are repeated twice then it’s solution is
((A)) 𝑒 𝛽𝑥 [𝑐1 cos𝛼𝑥 + 𝑐2 sin𝛼𝑥]
((B)) 𝑒 𝛼𝑥 [(𝑐1 𝑥 + 𝑐2 )cos𝛽𝑥 + (𝑐3 𝑥 + 𝑐4 )sin𝛽𝑥]
((C)) (𝑐1 𝑥 + 𝑐2 )𝑒 𝛼𝑥 + (𝑐3 𝑥 + 𝑐4 )𝑒 𝛽𝑥
((D)) 𝑒 𝛼𝑥 [𝑐1 cos𝛽𝑥 + 𝑐2 sin𝛽𝑥 ]
((E))B
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 − 5 𝑑𝑥 + 6𝑦 = 0 is
((A)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 −3𝑥
((B)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 3𝑥
((C)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 −3𝑥
((D)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 3𝑥
((E))D
((F))
 𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of DE − 5 𝑑𝑥 − 6𝑦 = 0 is
𝑑𝑥 2
((A)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 6𝑥
((B)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 − 3𝑥
((C)) 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 2𝑥
((D)) 𝑐1 𝑒 −3𝑥 + 𝑐2 𝑒 − 2𝑥
((E))A
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E. 2 𝑑𝑥 2 − 𝑑𝑥 − 10𝑦 = 0 is
5
((A)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 ( ⁄2)𝑥
5
((B)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 −( ⁄2)𝑥
5
((C)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 ( ⁄2)𝑥
3
((D)) 𝑐1 𝑒 −2𝑥 + 𝑐2 𝑒 ( ⁄2)𝑥
((E))A
((F))

𝑑2𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 − 4𝑦 = 0 is
((A)) (𝑐1 𝑥 + 𝑐2 )𝑒 2𝑥
((B)) 𝑐1 𝑒 4𝑥 + 𝑐2 𝑒 −4𝑥
((C)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 −2𝑥
((D)) 𝑐1 cos2𝑥 + 𝑐2 sin2𝑥
((E))C
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 + 2 𝑑𝑥 + 𝑦 = 0 is
((A)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 𝑥
((B)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )𝑒 −𝑥
((D)) (𝑐1 𝑥 + 𝑐2 )𝑒 𝑥
((E))C
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 − 4 𝑑𝑥 + 4𝑦 = 0 is
((A)) (𝑐1 𝑥 + 𝑐2 )𝑒 2𝑥
((B)) (𝑐1 𝑥 + 𝑐2 )𝑒 −2𝑥
((C)) 𝑐1 𝑒 4𝑥 + 𝑐2 𝑒 −4𝑥
((D)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 −2𝑥
((E))A
((F))
 𝑑3𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 3 − 4 𝑑𝑥 = 0 is
((A)) 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 −2𝑥
((B)) 𝑐1 + 𝑐2 cos2𝑥 + 𝑐3 sin2𝑥
((C)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 −3𝑥
((D)) 𝑐1 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 −2𝑥
((E))D
((F))

((Q))1_// The roots 𝑚1 , 𝑚2 , 𝑚3 ⋯ ⋯ , 𝑚𝑛 of auxiliary equation 𝜙(𝐷) = 0 are real. If three of these roots are
repeated, s((A))y, 𝑚1 = 𝑚2 = 𝑚3 and the remaining roots 𝑚4 , 𝑚5 ⋯ ⋯ 𝑚𝑛 are distinct then solution of
𝜙(𝐷)𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑚1𝑥 + 𝑐2 𝑒 𝑚2𝑥 + ⋯ ⋯ + 𝑐𝑛 𝑒 𝑚𝑛𝑥
((B)) (𝑐1 𝑥 2 + 𝑐2 𝑥 + 𝑐3 )𝑒 𝑚1𝑥 + 𝑐4 𝑒 𝑚4𝑥 + ⋯ ⋯ + 𝑐𝑛 𝑒 𝑚𝑛𝑥
((C)) (𝑐1 𝑥 2 + 𝑐2 𝑥 + 𝑐3 )cos𝑚1 𝑥 + 𝑐4 cos𝑚4 𝑥 + ⋯ ⋯ + 𝑐𝑛 cos𝑚𝑛 𝑥
((D)) (𝑐1 𝑥 2 + 𝑐2 𝑥 + 𝑐3 )sin𝑚1 𝑥 + 𝑐4 sin𝑚4 𝑥 + ⋯ ⋯ + 𝑐𝑛 sin𝑚𝑛 𝑥
((E))B
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 + 6 𝑑𝑥 + 9𝑦 = 0 is
((A)) 𝑐1 𝑒 −6𝑥 + 𝑐2 𝑒 −9𝑥
((B)) (𝑐1 𝑥 + 𝑐2 )𝑒 −3𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )𝑒 3𝑥
((D)) 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 2𝑥
((E))B
((F))

𝑑2𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 + 𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥
((B)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )𝑒 −𝑥
((D)) 𝑒 𝑥 (𝑐1 cos𝑥 + 𝑐2 sin𝑥)
((E))B
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 + 6 𝑑𝑥 + 10𝑦 = 0 is
((A)) 𝑒 𝑥 (𝑐1 cos𝑥 + 𝑐2 sin𝑥)
((B)) 𝑒 𝑥 (𝑐1 cos3𝑥 + 𝑐2 sin3𝑥)
((C)) 𝑐1 𝑒 5𝑥 + 𝑐2 𝑒 2𝑥
((D)) 𝑒 −3𝑥 (𝑐1cos𝑥 + 𝑐2 sin𝑥)
((E))D
((F))
 𝑑2𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 2 + 𝑑𝑥 + 𝑦 = 0 is
((A)) 𝑒 𝑥 (𝑐1 cos𝑥 + 𝑐2 sin𝑥)
x⁄ 3 3
((B)) 𝑒 2 [𝑐1 cos ( ) 𝑥 + 𝑐2 sin ( ) 𝑥]
2 2
−𝑥⁄2 √3 √3
((C)) 𝑒 [𝑐1 cos ( 2 ) 𝑥 + 𝑐2 sin ( 2 ) 𝑥]
((D)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥
((E))C
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 3 − 2 𝑑𝑥 2 + 4 𝑑𝑥 − 8y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((B)) 𝑦 = 𝑐1 𝑒 −2x + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 2x + 𝑐3 𝑒 −2x
((D)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −2x + 𝑐3 𝑒 2x
((E))A
((F))

𝑑3𝑦 𝑑𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 3 − 2 𝑑𝑥 + 4y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 2x + 𝑒 𝑥 [𝑐2 cos(𝑥) + 𝑐3 sin(𝑥)]
((B)) 𝑦 = 𝑐1 𝑒 2x + 𝑒 −𝑥 [𝑐2 cos(𝑥) + 𝑐3 sin(𝑥)]
((C)) 𝑦 = 𝑐1 𝑒 2x + 𝑐2 𝑒 −2x + 𝑐3 𝑒 𝑥
((D)) 𝑦 = 𝑐1 𝑒 2x + 𝑐2 𝑒 −𝑥 + 𝑐3 𝑒 𝑥
((E))B
((F))

𝑑4𝑦 𝑑2𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 4 + 5 𝑑𝑥 2 + 4y = 0 is
((A)) 𝑦 = 𝑐1 cos(𝑥) + 𝑐2 sin(𝑥) + 𝑐3 𝑒 2x + 𝑐4 𝑒 −2x
((B)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 + 𝑐3 𝑒 2x + 𝑐4 𝑒 −2x
((C)) 𝑦 = 𝑐1 cos(𝑥) + 𝑐2 sin(𝑥) + 𝑐3 cos(2x) + 𝑐4 sin(2x)
((D)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 + 𝑐3 cos(2x) + 𝑐4 sin(2x)
((E))C
((F))

𝑑3𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 3 + 𝑦 = 0 is
((A)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑒 𝑥 [𝑐2 cos(√3𝑥) + 𝑐3 sin(√3𝑥)]
((B)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 𝑥/2 + 𝑐3 𝑒 √3𝑥/2
√3𝑥 √3𝑥
((C)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑒 −𝑥/2 [𝑐2 cos(
) + 𝑐3 sin( )]
2 2
−𝑥 𝑥/2 √3𝑥 √3𝑥
((D)) 𝑦 = 𝑐1 𝑒 + 𝑒 [𝑐2cos( 2 ) + 𝑐3 sin( 2 )]
((E))D
((F))
 𝑑3𝑦 𝑑𝑦
((Q))1_// A solution of D.E.4 𝑑𝑥 3 − 7 𝑑𝑥 + 3y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 2x + 𝑐2 𝑒 𝑥/2 + 𝑐3 𝑒 −𝑥/2
((B)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 𝑥/2 + 𝑐3 𝑒 −3x/2
((C)) 𝑦 = 𝑐1 + 𝑐2 𝑒 −𝑥 + 𝑐3 𝑒 −3x/4
((D)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 𝑥/2 + 𝑐3 𝑒 −3x/2
((E))B
((F))

((Q))1_// If the roots 𝑚1 , 𝑚2 , 𝑚3 ⋯ ⋯ , 𝑚𝑛 of Auxiliary equation 𝜙(𝐷) = 0 Are real And distinct, then solution
of 𝜙(𝐷)𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑚1𝑥 + 𝑐2 𝑒 𝑚2𝑥 + ⋯ ⋯ + 𝑐𝑛 𝑒 𝑚𝑛𝑥
((B)) 𝑐1 cos𝑚1 𝑥 + 𝑐2 cos𝑚2 𝑥 + ⋯ ⋯ + 𝑐𝑛 cos𝑚𝑛 𝑥
((C)) 𝑚1 𝑒 𝑐1𝑥 + 𝑚2 𝑒 𝑐2𝑥 + ⋯ ⋯ + 𝑚𝑛 𝑒 𝑐𝑛𝑥
((D)) 𝑐1 sin𝑚1 𝑥 + 𝑐2 sin𝑚2 𝑥 + ⋯ ⋯ + 𝑐𝑛 sin𝑚𝑛 𝑥
((E))A
((F))

𝑑3𝑦 𝑑𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 3 − 3 𝑑𝑥 + 2y = 0 is
((A)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 𝑥 + 𝑐3 𝑒 −2x
((B)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −𝑥 + 𝑐3 𝑒 2x
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 2x + 𝑐3 𝑒 −2x
((D)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2x + 𝑐3 𝑒 3x
((E))A
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// A solution of D.E. 𝑑𝑥 3 − 𝑑𝑥 2 + 3 𝑑𝑥 + 5y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑒 𝑥 [𝑐2 cos(2x) + 𝑐3 sin(2x)]
((B)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑒 −𝑥 [𝑐2 cos(2x) + 𝑐3 sin(2x)]
((C)) 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑒 −𝑥 [𝑐2 cos(2x) + 𝑐3 sin(2x)]
((D)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑒 𝑥 [𝑐2 cos(2x) + 𝑐3 sin(2x)]
((E))A
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))2_// A solution of D.E. 𝑑𝑥 3 − 5 𝑑𝑥 2 + 7 𝑑𝑥 − 3y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((B)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 𝑥 + 𝑐3 𝑒 3x
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −𝑥 + 𝑐3 𝑒 −3x
((D)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −3x + 𝑐3 𝑒 𝑥
((E)) B
((F))
 𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))2_// A solution of D.E. 𝑑𝑥 3 − 6 𝑑𝑥 2 + 12 𝑑𝑥 − 8y = 0 is
((A)) 𝑦 = (𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑥 2 )𝑒 2x
((B)) 𝑦 = 𝑐1 𝑒 −2x + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 2x + 𝑐3 𝑒 −2x
((D)) 𝑦 = (𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑥 2 )𝑒 −2x
((E)) A
((F))

𝑑4𝑦 𝑑2𝑦
((Q))2_// A solution of D.E. 𝑑𝑥 4 + 18 𝑑𝑥 2 + 81y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 3x + 𝑐2 𝑒 −3x + 𝑐3 cos(3x) + 𝑐4 sin(3x)
((B)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −3x + 𝑐3 cos(2x) + 𝑐4 sin(2x)
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 3x + (𝑐3 + 𝑐4 𝑥)𝑒 −3x
((D)) 𝑦 = (𝑐1 + 𝑐2 𝑥)cos(3x) + (𝑐3 + 𝑐4 𝑥)sin(3x)
((E)) D
((F))

𝑑3𝑦 𝑑𝑦
((Q))2_// A solution of D.E. 𝑑𝑥 3 − 12 𝑑𝑥 + 16y = 0 is
((A)) 𝑦 = 𝑐1 𝑒 4x + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((B)) 𝑦 = 𝑐1 𝑒 −2x + 𝑐2 cos(2x) + 𝑐3 sin(2x)
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 2x + 𝑐3 𝑒 −4x
((D)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 −4x + 𝑐3 𝑒 2x
((E)) C
((F))

𝑑4𝑦 𝑑3𝑦 𝑑2𝑦

((Q))1_// A solution of D.E. 𝑑𝑥 4 + 𝑑𝑥 3 + 𝑑𝑥 2 = 0 is
√3 √3
((A)) 𝑦 = 𝑐1 + 𝑐2 𝑥 + 𝑒 −𝑥/2 [𝑐3 cos( 2 𝑥) + 𝑐4 sin( 2 𝑥)]
((B)) 𝑦 = 𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑒 𝑥 + 𝑐4 𝑒 −𝑥
((C)) 𝑦 = (𝑐1 + 𝑐2 𝑥)𝑒 𝑥 + (𝑐3 + 𝑐4 𝑥)𝑒 −𝑥
((D)) 𝑦 = 𝑐1 + 𝑐2 𝑥 + 𝑒 −𝑥 [𝑐3cos(√3𝑥) + 𝑐4 sin(√3𝑥)]
((E)) A
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// Solution of D.E.𝑑𝑥 3 + 6 𝑑𝑥 2 + 11 𝑑𝑥 + 6𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 3𝑥
((B)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 −3𝑥
((C)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 −3𝑥
((D)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 3𝑥
((E)) C
((F))
 𝑑3𝑦 𝑑𝑦
((Q))1_// Solution of D.E.𝑑𝑥 3 − 7 𝑑𝑥 − 6𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 3𝑥
((B)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 6𝑥
((C)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 𝑥
((D)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 3𝑥
((E)) D
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// Solution of D.E.𝑑𝑥 3 + 2 𝑑𝑥 2 + 𝑑𝑥 = 0 is
((A)) 𝑐1 + 𝑐2 𝑒 𝑥 + 𝑐3 𝑒 −𝑥
((B)) 𝑐1 + 𝑒 𝑥 (𝑐2 𝑥 + 𝑐3 )
((C)) 𝑒 −𝑥 (𝑐2 𝑥 + 𝑐3 )
((D)) 𝑐1 + 𝑒 −𝑥 (𝑐2 𝑥 + 𝑐3 )
((E)) D
((F))

𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// Solution of D.E.𝑑𝑥 3 − 5 𝑑𝑥 2 + 8 𝑑𝑥 − 4𝑦 = 0 is
((A)) 𝑐1 𝑒 𝑥 + (𝑐2 𝑥 + 𝑐3 )𝑒 2𝑥
((B)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 3𝑥
((C)) (𝑐2 𝑥 + 𝑐3 )𝑒 2𝑥
((D)) 𝑐1 𝑒 −𝑥 + (𝑐2 𝑥 + 𝑐3 )𝑒 −2𝑥
((E)) A
((F))

𝑑3𝑦
((Q))2_// The solution of D.E.𝑑𝑥 3 + 𝑦 = 0 is
√3 √3
((A)) 𝑐1 𝑒 𝑥 + 𝑒 𝑥 (𝑐2 cos 𝑥 + 𝑐3 sin 𝑥)
2 2
1
1 1
((B)) 𝑐1 𝑒 − 𝑥 + 𝑒 2 𝑥 (𝑐2 cos 2 𝑥 + 𝑐3 sin 2 𝑥)
1
√3 √3
((C)) 𝑐1 𝑒 − 𝑥 + 𝑒 2 𝑥 (𝑐2 cos 2 𝑥 + 𝑐3 sin 2
𝑥)
((D)) (𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑥 2 )𝑒 − 𝑥
((E)) C
((F))

𝑑3𝑦 𝑑𝑦
((Q))1_// The solution of D.E.𝑑𝑥 3 + 3 𝑑𝑥 = 0 is
((A)) 𝑐1 + 𝑐2 cos𝑥 + 𝑐3 sin𝑥
((B)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥
((C)) 𝑐1 + 𝑐2 𝑒 √3 𝑥 + 𝑐3 𝑒 −√3 𝑥
((D)) 𝑐1 + 𝑐2 cos√3𝑥 + 𝑐3 sin√3𝑥
((E)) D
((F))
 𝑑3𝑦 𝑑2𝑦 𝑑𝑦
((Q))1_// Solution of D.E.𝑑𝑥 3 + 𝑑𝑥 2 − 2 𝑑𝑥 + 12𝑦 = 0 is
((A)) 𝑐1 𝑒 −3𝑥 + 𝑒 𝑥 (𝑐2 cos√3𝑥 + 𝑐3 sin√3𝑥)
((B)) 𝑐1 𝑒 −3𝑥 + (𝑐2 cos3𝑥 + 𝑐3 sin3𝑥)
((C)) 𝑐1 𝑒 3𝑥 + 𝑒 −𝑥 (𝑐2cos√3𝑥 + 𝑐3 sin√3𝑥)
((D)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −√3 𝑥 + 𝑐3 𝑒 √3 𝑥
((E)) A
((F))

((Q))1_// Solution of D.E.(𝐷3 − 𝐷2 + 3𝐷 + 5)𝑦 = 0 is

((A)) 𝑐1 𝑒 −𝑥 + 𝑒 𝑥 (𝑐2 cos2𝑥 + 𝑐3 sin2𝑥)
((B)) 𝑐1 𝑒 −𝑥 + (𝑐2 cos3𝑥 + 𝑐3 sin3𝑥)
((C)) 𝑐1 𝑒 𝑥 + 𝑒 −𝑥 (𝑐2 cos2𝑥 + 𝑐3 sin2𝑥)
((D)) 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 −3𝑥
((E)) A
((F))

𝑑4𝑦
((Q))2_// The solution of D.E.𝑑𝑥 4 − 𝑦 = 0 is
((A)) (𝑐1 𝑥 + 𝑐2 )𝑒 −𝑥 + 𝑐3 cos𝑥 + 𝑐4 sin𝑥
((B)) (𝑐1 𝑥 + 𝑐2 )cos𝑥 + (𝑐3 𝑥 + 𝑐4 )sin𝑥
((C)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 − 𝑥 + 𝑐3 cos𝑥 + 𝑐4 sin𝑥
((D)) (𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑥 2 + 𝑐4 𝑥 3 )𝑒 𝑥
((E)) C
((F))

((Q))2_// The solution of D.E.(𝐷4 + 2𝐷2 + 1)𝑦 = 0 is

((A)) (𝑐1 𝑥 + 𝑐2 )𝑒 𝑥 + (𝑐3 𝑥 + 𝑐4 )𝑒 −𝑥
((B)) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 + 𝑐3 cos𝑥 + 𝑐4 sin𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )cos𝑥 + (𝑐3 𝑥 + 𝑐4 )sin𝑥
((D)) (𝑐1 𝑥 + 𝑐2 )cos2𝑥 + (𝑐3 𝑥 + 𝑐4 )sin2𝑥
((E)) C
((F))

((Q))2_// The solution of D.E.(𝐷2 + 9)2 𝑦 = 0 is

((A)) (𝑐1 𝑥 + 𝑐2 )𝑒 3𝑥 + (𝑐3 𝑥 + 𝑐4 )𝑒 −3𝑥
((B)) (𝑐1 𝑥 + 𝑐2 )cos3𝑥 + (𝑐3 𝑥 + 𝑐4 )sin3𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )cos9𝑥 + (𝑐3 𝑥 + 𝑐4 )sin9𝑥
((D)) (𝑐1 𝑥 + 𝑐2 )cos𝑥 + (𝑐3 𝑥 + 𝑐4 )sin𝑥
((E)) B
((F))

𝑑6𝑦 𝑑4𝑦 𝑑2𝑦

((Q))2_// The solution of D.E.𝑑𝑥 6 + 6 𝑑𝑥 4 + 9 𝑑𝑥 2 = 0 is
 ((A)) 𝑐1 𝑥 + 𝑐2 + (𝑐3 𝑥 + 𝑐4 )cos√3 𝑥 + (𝑐5 𝑥 + 𝑐6 )sin√3 𝑥
((B)) 𝑐1 𝑥 + 𝑐2 + (𝑐3 𝑥 + 𝑐4 )cos3 𝑥 + (𝑐5 𝑥 + 𝑐6 )sin3 𝑥
((C)) (𝑐1 𝑥 + 𝑐2 )cos√3 𝑥 + (𝑐3 𝑥 + 𝑐4 )sin√3 𝑥
((D)) 𝑐1 𝑥 + 𝑐2 + (𝑐3 𝑥 + 𝑐4 )𝑒 √3𝑥
((E)) A
((F))

1 𝑑
((Q))1_// 𝐷+𝑚 𝑓(𝑥) , where 𝐷 ≡ 𝑑𝑥 and 𝑚 is constant, is equal to
((A)) 𝑒 −𝑚𝑥 ∫ 𝑒 𝑚𝑥 𝑑𝑥
((B)) ∫ 𝑒 𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((C)) 𝑒 𝑚𝑥 ∫ 𝑒 −𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((D)) 𝑒 −𝑚𝑥 ∫ 𝑒 𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((E)) D
((F))

1
((Q))1_//Particular Integral of 𝜙(𝐷2) sin(𝑎𝑥 + 𝑏),𝜙(−𝑎2 ) = 0 , 𝜙 ′ (−𝑎2 ) ≠ 0is
1
((A)) 𝑥 𝜙′ (−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((B)) 𝑥 𝜙′(−𝑎2) sin(𝑎𝑥 + 𝑏)
1
((C)) 𝜙(−𝑎2) sin(𝑎𝑥 + 𝑏)
1
((D)) 𝜙′(−𝑎2) sin(𝑎𝑥 + 𝑏)
((E)) B
((F))

((Q))1_// Particular Integral of D.E.(𝐷2 + 1)𝑦 = sin𝑥 is

𝑥
((A)) − 2 cos𝑥
𝑥
((B)) − 4 cos𝑥
𝑥
((C)) − 2 sin𝑥
1
((D)) − 2 cos𝑥
((E)) A
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷) 𝑒 𝑎 𝑥 𝑉 where 𝑉 is any function of 𝑥 is
1
((A)) 𝑒 𝑎𝑥 𝜙(𝐷−𝑎) 𝑉
1
((B)) 𝑒 𝑎𝑥 𝜙(𝑎) 𝑉
1
((C)) 𝑒 𝑎 𝑥 𝑉
𝜙(𝐷+𝑎)
1
((D)) 𝜙(𝐷+𝑎) 𝑉
((E)) C
((F))
((Q))2_// Particular Integral of D.E.(𝐷3 + 9𝐷)𝑦 = sin3𝑥 is
𝑥
((A)) − 18 cos3𝑥
𝑥
((B)) − 18 sin3𝑥
((C)) −𝑥sin3𝑥
1
((D)) − 18 sin3𝑥
((E)) B
((F))

1
((Q))1_// Particular Integral of (𝐷−𝑎)𝑟 𝑒 𝑎 𝑥 is
1
((A)) 𝑟! 𝑒 𝑎𝑥
𝑥𝑟
((B)) 𝑒 𝑎𝑥
𝑟
𝑥𝑟
((C)) 𝑟! 𝑒 𝑎𝑥
((D)) 𝑥 𝑟 𝑒 𝑎𝑥
((E)) C
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷2) sin(𝑎𝑥 + 𝑏), 𝜙(−𝑎2 ) ≠ 0 is
1
((A)) 𝜙(−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((B)) 𝜙(−𝑎2) sin(𝑎𝑥 + 𝑏)
1
((C)) 𝑥 𝜙′(−𝑎2) sin(𝑎𝑥 + 𝑏)
1
((D)) 𝜙(𝑎2) sin(𝑎𝑥 + 𝑏)
((E)) B
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷2) cos(𝑎𝑥 + 𝑏), 𝜙(−𝑎2 ) ≠ 0 is
1
((A)) 𝜙(−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((B)) sin(𝑎𝑥 + 𝑏)
𝜙(−𝑎2 )
1
((C)) 𝑥 𝜙′(−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((D)) 𝜙(𝑎2) cos(𝑎𝑥 + 𝑏)
((E)) A
((F))

((Q))2_// The Particular Integral of D.E.(𝐷4 + 10𝐷2 + 9)𝑦 = sin2𝑥 + cos4𝑥 is

1 1
((A)) − 5 sin2𝑥 − 29 cos4𝑥
1
((B)) 15 sin2𝑥 + cos4𝑥
 1 1
((C)) − 15 cos4𝑥 − 87 sin2𝑥
1 1
((D)) − 15 sin2𝑥 − 87 cos4𝑥
((E)) C
((F))

𝑑2𝑦 𝑑𝑦
((Q))2_// Particular Integral of D.E.𝑑𝑥 2 − 2 𝑑𝑥 + 5𝑦 = 10sin𝑥 is
8
((A)) 3 sin𝑥
((B)) sin𝑥 − 2cos𝑥
((C)) 4sin𝑥 + 2cos𝑥
((D)) 2sin𝑥 + cos𝑥
((E)) D
((F))

1 𝑑
((Q))1_// 𝐷−𝑚 𝑓(𝑥) , where 𝐷 ≡ 𝑑𝑥 And 𝑚 is constant, is
((A)) 𝑒 𝑚𝑥 ∫ 𝑒 −𝑚𝑥 𝑑𝑥
((B)) ∫ 𝑒 −𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((C)) 𝑒 𝑚𝑥 ∫ 𝑒 −𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((D)) 𝑒 −𝑚𝑥 ∫ 𝑒 𝑚𝑥 𝑓(𝑥) 𝑑𝑥
((E)) C
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷) 𝑒 𝑎 𝑥 , 𝜙(𝑎) ≠ 0 is
1
((A)) 𝜙(−𝑎) 𝑒 𝑎𝑥
1
((B)) 𝑥 𝜙′(𝑎) 𝑒 𝑎𝑥
1
((C)) 𝜙(𝑎2) 𝑒 𝑎𝑥
1
((D)) 𝑒 𝑎𝑥
𝜙(𝑎)
((E)) D
((F))

1
((Q))1_// Particular Integral for 𝜙(𝐷) 𝑥𝑉 , where 𝑉 is ((A)) function of 𝑥,is
1 1
((A)) [𝑥 − ] 𝑉
𝜙(𝐷) 𝜙(𝐷)
𝜙 ′ (𝐷)
((B)) [𝑥 − ] 𝜙(𝐷)𝑉
𝜙(𝐷)
𝜙 ′ (𝐷)
((C)) [𝑥 + ] 𝑉
𝜙(𝐷)
𝜙 ′ (𝐷) 1
((D)) [𝑥 − ] 𝑉
𝜙(𝐷) 𝜙(𝐷)
((E)) D
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷2) cos(𝑎𝑥 + 𝑏), 𝜙(−𝑎2 ) = 0, 𝜙′ (−𝑎2 ) ≠ 0 is
1
((A)) 𝜙′ (−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((B)) 𝜙′(−𝑎2) cos(𝑎𝑥 + 𝑏)
1
((C)) 𝑥 𝜙′(−𝑎2) sin(𝑎𝑥 + 𝑏)
1
((D)) 𝑥 𝜙′(−𝑎2) cos(𝑎𝑥 + 𝑏)
((E)) D
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷2) sinh(𝑎𝑥 + 𝑏), 𝜙(𝑎2 ) ≠ 0 is
1
((A)) 𝜙(𝑎2) cosh(𝑎𝑥 + 𝑏)
1
((B)) 𝑥 𝜙′(𝑎2) sinh(𝑎𝑥 + 𝑏)
1
((C)) 𝜙(𝑎2) sinh(𝑎𝑥 + 𝑏)
1
((D)) 𝜙(−𝑎2) sinh(𝑎𝑥 + 𝑏)
((E)) C
((F))

1
((Q))1_// Particular Integral of 𝜙(𝐷2) cosh(𝑎𝑥 + 𝑏),𝜙(𝑎2 ) ≠ 0 , is
1
((A)) 𝜙(𝑎2) cosh(𝑎𝑥 + 𝑏)
1
((B)) 𝑥 𝜙′(𝑎2) cosh(𝑎𝑥 + 𝑏)
1
((C)) 𝜙(𝑎2) sinh(𝑎𝑥 + 𝑏)
1
((D)) 𝜙(−𝑎2) cosh(𝑎𝑥 + 𝑏)
((E)) A
((F))

1 𝑥 𝑑
((Q))1_// Particular Integral𝐷+1 𝑒 𝑒 where 𝐷 ≡ 𝑑𝑥 is
𝑥
((A)) 𝑒 −𝑥 𝑒 𝑒
𝑥
((B)) 𝑒 𝑒
𝑥
((C)) 𝑒 𝑥 𝑒 𝑒
𝑥
((D)) 𝑒 − 2𝑥 𝑒 𝑒
((E)) A
((F))
 1 𝑥 𝑑
((Q))2_// Particular Integral𝐷+2 𝑒 −𝑥 𝑒 𝑒 where 𝐷 ≡ 𝑑𝑥 is
𝑥
((A)) 𝑒 2𝑥 𝑒 𝑒
𝑥
((B)) 𝑒 − 2𝑥 𝑒 𝑒
𝑥
((C)) 𝑒 𝑒
𝑥
((D)) 𝑒 − 𝑥 𝑒 𝑒
((E)) B
((F))

1 𝑑
((Q))2_// Particular Integral𝐷+2 𝑒 −𝑥 cos𝑒 𝑥where 𝐷 ≡ 𝑑𝑥 is
((A)) 𝑒 −𝑥 cos𝑒 𝑥
((B)) 𝑒 −𝑥 sin𝑒 𝑥
((C)) 𝑒 −2𝑥 cos𝑒 𝑥
((D)) 𝑒 −2𝑥 sin𝑒 𝑥
((E)) D
((F))

((Q))1_// Particular Integral of D.E. (𝐷2 − 9)𝑦 = 𝑒 3𝑥 + 1 is

3𝑥 1
((A)) 2 𝑒 3𝑥 − 8
𝑒 3𝑥 3
((B)) 𝑥 +8
6
𝑒 3𝑥 1
((C)) 𝑥 −8
6
3𝑥 1
((D)) 𝑥𝑒 +8
((E)) C
((F))

((Q))2_// Particular Integral of D.E.(𝐷3 + 𝐷)𝑦 = cos𝑥 is

𝑥
((A)) − 2 sin𝑥
𝑥
((B)) cos𝑥
4
1
((C)) − 2 cos𝑥
𝑥
((D)) − 2 cos𝑥
((E)) D
((F))

1 𝑑
((Q))2_// Particular Integral𝐷+2 𝑒 −2𝑥 sec 2 𝑥(1 + 2t((A))n𝑥) where 𝐷 ≡ 𝑑𝑥 is
((A)) 𝑒 −2𝑥 (1 + 2t((A))n2 𝑥)
((B)) 𝑒 −2𝑥 (t((A))n𝑥 + t((A))n2 𝑥)
((C)) 𝑒 2𝑥 (t((A))n𝑥 + 2t((A))n2 𝑥)
((D)) 𝑒 −2𝑥 (t((A))n𝑥 + sec𝑥)
((E)) B
((F))
 1 1 𝑑
((Q))2_// Particular Integral𝐷+1 (1+𝑒 𝑥)where 𝐷 ≡ 𝑑𝑥 is
((A)) 𝑒 𝑥 log(1 − 𝑒 𝑥 )
((B)) log(1 + 𝑒 𝑥 )
((C)) 𝑒 𝑥 log(1 + 𝑒 𝑥 )
((D)) 𝑒 −𝑥 log(1 + 𝑒 𝑥 )
((E)) D
((F))

𝑑2𝑦 𝑑𝑦
((Q))2_// The Particular Integral of D.E.𝑑𝑥 2 − 7 𝑑𝑥 + 6𝑦 = 𝑒 2𝑥 is
𝑥𝑒 2𝑥
((A)) − 3
𝑒 2𝑥
((B)) − 4
𝑒 2𝑥
((C)) 4
𝑒 2𝑥
((D))
24
((E)) B
((F))

1 𝑑
((Q))2_// Particular Integral sin𝑒 𝑥 where 𝐷 ≡ is
𝐷+1 𝑑𝑥
((A)) − 𝑒 −𝑥 sin𝑒 𝑥
((B)) 𝑒 𝑥 cos𝑒 𝑥
((C)) − 𝑒 −𝑥 cos𝑒 𝑥
((D)) 𝑒 −𝑥 cos𝑒 𝑥
((E)) C
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 − 5𝐷 + 6)𝑦 = 3𝑒 5𝑥 is

𝑒 5𝑥
((A)) 2
𝑒 5𝑥
((B)) 6
𝑒 5𝑥
((C)) − 14
𝑒 2𝑥
((D)) − 2
((E)) A
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 + 4𝐷 + 3)𝑦 = 𝑒 −3𝑥 is

((A)) 𝑥𝑒 −3𝑥
1
((B)) − 2 𝑒 −3𝑥
𝑥
((C)) − 10 𝑒 −3𝑥
𝑥
((D)) − 2 𝑒 −3𝑥
((E)) D
((F))
((Q))2_// The Particular Integral of D.E.(𝐷 − 2)3 𝑦 = 𝑒 2𝑥 + 3𝑥 is
𝑥3 1
((A)) 𝑒 2𝑥 + (log3−2)3 3𝑥
3!
𝑥3 1
((B)) 𝑒 2𝑥 + (𝑒 3 3 3𝑥
3! −2)
𝑥 1
((C)) 3! 𝑒 2𝑥 + (log3−2)3 3𝑥
𝑥3 1
((D)) 3! 𝑒 2𝑥 + (log3−2)3
((E)) A
((F))

((Q))2_// The Particular Integral of D.E.(𝐷5 − 𝐷)𝑦 = 12𝑒 𝑥 is

((A)) 3 𝑒 𝑥
12
((B)) 5 𝑥 𝑒 𝑥
((C)) 12𝑥 𝑒 𝑥
((D)) 3𝑥 𝑒 𝑥
((E)) D
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 + 1)(𝐷 − 1)𝑦 = 𝑒 𝑥 is

((A)) 𝑥 𝑒 𝑥
1
((B)) 2 𝑥 2 𝑒 𝑥
1
((C)) 2 𝑥 𝑒 𝑥
((D)) 𝑥 2 𝑒 𝑥
((E)) C
((F))

((Q))1_// The Particular Integral of D.E.(𝐷2 − 4𝐷 + 4)𝑦 = sin2𝑥is

cos2𝑥
((A)) − 8
cos2𝑥
((B)) 8
sin2𝑥
((C)) 8
cos2𝑥
((D)) 𝑥 8
((E)) B
((F))

((Q))1_// The Particular Integral of D.E.(𝐷4 − 𝑚4 )𝑦 = cos𝑚𝑥 is

−𝑥
((A)) 3 cos𝑚𝑥
4𝑚
𝑥
((B)) 𝑚3 sin𝑚𝑥
((C)) −𝑥sin𝑚𝑥
−𝑥
((D)) 4𝑚3 sin𝑚𝑥
((E)) D
((F))

𝑑3𝑦 𝑑𝑦
((Q))2_// The Particular Integral of D.E.𝑑𝑥 3 − 4 𝑑𝑥 = 2cosh2𝑥 is
1
((A)) 4 cosh2𝑥
𝑥
((B)) 8 cosh2𝑥
𝑥
((C)) 4 cosh2𝑥
𝑥
((D)) 4 sinh2𝑥
((E)) C
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 + 6𝐷 − 9)𝑦 = sinh3𝑥 is

1
((A)) 18 cosh3𝑥
1
((B)) 2 cosh3𝑥
1
((C)) 18 sinh3𝑥
1
((D)) − 18 cosh3𝑥
((E)) A
((F))

𝑑3𝑦
((Q))2_// The Particular Integral of D.E.𝑑𝑥 3 + 8𝑦 = 𝑥 4 + 2𝑥 + 1 is
1
((A)) 8 (𝑥 4 + 5𝑥 + 1)
1
((B)) 8 (𝑥 3 − 3𝑥 2 + 1)
((C)) 𝑥 4 − 𝑥 + 1
1
((D)) 8 (𝑥 4 − 𝑥 + 1)
((E)) D
((F))

((Q))2_// The Particular Integral of D.E.(𝐷4 + 𝐷2 + 1)𝑦 = 53𝑥 2 + 17 is

((A)) 53𝑥 2 + 17
((B)) 53𝑥 2 − 89
((C)) 53𝑥 2 + 113
((D)) 3𝑥 2 − 17
((E)) B
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 − 𝐷 + 1)𝑦 = 𝑥 3 − 3𝑥 2 + 1 is

((A)) 𝑥 3 + 6𝑥 + 5
((B)) 𝑥 2 − 6𝑥 + 1
((C)) 𝑥 3 − 6𝑥 − 5
((D)) 𝑥 3 − 6𝑥 2 + 18𝑥 − 11
((E)) C
((F))

((Q))1_// The Particular Integral of D.E.(𝐷2 − 1)𝑦 = 𝑥 3 is

((A)) −𝑥 3 + 6𝑥
((B)) 𝑥 2 + 6
((C)) 𝑥 3 + 6𝑥
((D)) −𝑥 3 − 6𝑥
((E)) D
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 + 2𝐷 + 1)𝑦 = 𝑒 −𝑥 (1 + 𝑥 2 )is

𝑥2 𝑥4
((A)) 𝑒 𝑥 ( 2 − 12)
𝑥3
((B)) 𝑒 − 𝑥 (𝑥 + )
3
𝑥2 𝑥4
((C)) 𝑒 − 𝑥 ( 2 + 12)
𝑥2 𝑥4
((D)) ( 2 + 12)
((E)) C
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 − 4𝐷 + 4)𝑦 = 𝑒 2𝑥 𝑥 4 is

𝑥6
((A)) 120 𝑒 2𝑥
𝑥6
((B)) 60 𝑒 2𝑥
𝑥6
((C)) 30 𝑒 2𝑥
𝑥5
((D)) 20 𝑒 2𝑥
((E)) B
((F))

𝑑2𝑦 𝑑𝑦
((Q))2_// The Particular Integral of D.E.𝑑𝑥 2 + 2 𝑑𝑥 + 𝑦 = 𝑒 − 𝑥 cos𝑥 is
((A)) 𝑒 𝑥 cos𝑥
((B)) −𝑒 − 𝑥 sin𝑥
((C)) −𝑒 − 𝑥 cos𝑥
((D)) (𝑐1 𝑥 + 𝑐2 )𝑒 − 𝑥
((E)) C
((F))

((Q))2_// The Particular Integral of D.E.(𝐷2 + 6𝐷 + 9)𝑦 = 𝑒 −3𝑥 𝑥 −3 is

𝑒 − 3𝑥
((A)) 2𝑥
((B)) 𝑒 − 3𝑥 𝑥
𝑒 − 3𝑥
((C)) 12𝑥
((D)) (𝑐1 𝑥 + 𝑐2 )𝑒 − 3𝑥
((E)) A
((F))

((Q))2_// The Particular Integral of D.E.(𝐷3 + 3𝐷2 − 4)𝑦 = 𝑥 2 is

1 3
((A)) − 4 (𝑥 2 + 2)
1 3
((B)) 4 (𝑥 2 + 2 𝑥)
3
((C)) (𝑥 2 + 2)
1 3
((D)) − (𝑥 2 − )
4 2
((E)) A
((F))

((Q))2_// The Particular Integral of D.E.(𝐷 − 1)3 𝑦 = 𝑒 𝑥 √𝑥 is

4
((A)) 15 𝑒 𝑥 𝑥 5/2
8
((B)) 105 𝑒 𝑥 𝑥 7/2
((C)) 𝑒 𝑥 𝑥 7/2
3
((D)) 8 𝑒 𝑥 𝑥 − 5/2
((E)) B
((F))

((Q))2_// The Particular Integral of D.E.(𝐷4 + 25)𝑦 = 𝑥 4 + 𝑥 2 + 1 is

1
((A)) (𝑥 4 + 𝑥 2 − 25)
49
((B)) (𝑥 4 + 𝑥 2 + 25)
1
((C)) 25 (𝑥 4 + 𝑥 2 + 24𝑥 + 1)
1 1
((D)) 25 (𝑥 4 + 𝑥 2 + 25)
((E)) D
((F))

𝑑2𝑦 𝑑𝑦
((Q))2_// The Particular Integral of D.E.𝑑𝑥 2 − 2 + 𝑦 = 𝑥𝑒 𝑥 sin𝑥 is
𝑑𝑥
((A)) − 𝑒 𝑥 (𝑥sin𝑥 + 2cos𝑥)
((B)) 𝑒 𝑥 (𝑥sin𝑥 − 2cos𝑥)
((C)) (𝑥sin𝑥 + 2cos𝑥)
((D)) − 𝑒 𝑥 (𝑥cos𝑥 + 2sin𝑥)
((E)) A
((F))

𝑑2𝑦 𝑑𝑦
((Q))2_// Solution of D.E.𝑑𝑥 2 + 𝑑𝑥 + 𝑦 = 𝑒 2𝑥 is
√3 √3 1
((A)) 𝑒 𝑥 (𝑐1 cos 𝑥 + 𝑐2 sin 𝑥 ) − 7 𝑒 2𝑥
2 2
 1
√3 √3 1
((B)) 𝑒 2𝑥 (𝑐1 cos 𝑥 + 𝑐2 sin 𝑥 ) + 5 𝑒 2𝑥
2 2
1
1 1 1
((C)) 𝑒 − 2𝑥 (𝑐1 cos 2 𝑥 + 𝑐2 sin 2 𝑥 ) + 7 𝑒 𝑥
1
√3 √3 1
((D)) 𝑒 − 2𝑥 (𝑐1 cos 𝑥 + 𝑐2 sin 𝑥 ) + 7 𝑒 2𝑥
2 2
((E)) D
((F))

89. Solution of D.E.(𝐷2 + 1)𝑦 = 𝑥 is

((A)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥 + 𝑥
((B)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥 − 𝑥
((C)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥 + 2𝑥
((D)) 𝑐1 cos𝑥 + 𝑐2 sin𝑥 − 2𝑥
((E)) B
((F))

((Q))1_// The general form of Cauchy’s Linear equation is

−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((A)) 𝑎0 𝑑𝑥 𝑛 + 𝑎1 𝑑𝑥 𝑛−1 + 𝑎2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥), where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
𝑑𝑥 𝑑𝑦 𝑑𝑧
((B)) = = , where 𝑃, 𝑄, 𝑅 ((A))re function of 𝑥, 𝑦, 𝑧.
𝑃 𝑄 𝑅
𝑛𝑦 −1 −2
𝑛 𝑑 𝑛−1 𝑑𝑛 𝑦 𝑛−2 𝑑𝑛 𝑦
((C)) 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + 𝑎2 𝑥 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥),where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 Are constant
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑛−2
−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((D)) 𝑎0 (𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 𝑛 + 𝑎1 (𝑎𝑥 + 𝑏)𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 (𝑎𝑥 + 𝑏)𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥),where
𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
((E)) C
((F))

−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((Q))1_// Cauchy’s Linear equation 𝑎0 𝑥 𝑛 𝑑𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥) is reduced
to Linear D.E. with constant coefficient by using substitution
((A)) 𝑥 = 𝑒 𝑧
((B)) 𝑦 = 𝑒 𝑧
((C)) 𝑥 = log𝑧
2
((D)) 𝑥 = 𝑒 𝑧
((E)) A
((F))

((Q))1_// The general form of Legendre’s Linear equation is

−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((A)) 𝑎0 𝑑𝑥 𝑛 + 𝑎1 𝑑𝑥 𝑛−1 + 𝑎2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥), where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
𝑑𝑥 𝑑𝑦 𝑑𝑧
((B)) = = , where 𝑃, 𝑄, 𝑅 are function of 𝑥, 𝑦, 𝑧.
𝑃 𝑄 𝑅
𝑛𝑦 −1 −2
𝑛 𝑑 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((C)) 𝑎0 𝑥 + 𝑎1 𝑥 𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥),where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
𝑑𝑥 𝑛
−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((D)) 𝑎0 (𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 𝑛 + 𝑎1 (𝑎𝑥 + 𝑏)𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 (𝑎𝑥 + 𝑏)𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥), where
𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
((E)) D
((F))

−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((Q))1_// Legendre’s Linear differential 𝑎0 (𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 𝑛 + 𝑎1 (𝑎𝑥 + 𝑏)𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 (𝑎𝑥 + 𝑏)𝑛−2 𝑑𝑥 𝑛−2 +
⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥) is reduced to Linear D.E. with constant coefficient by using substitution
((A)) 𝑥 = 𝑒 𝑧
((B)) 𝑎𝑥 + 𝑏 = 𝑒 𝑧
((C)) 𝑎𝑥 + 𝑏 = log𝑧
2
((D)) 𝑎𝑥 + 𝑏 = 𝑒 𝑧
((E)) C
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// On putting 𝑥 = 𝑒 𝑧 the transformed D.E. of 𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 + 𝑦 = 𝑥is
𝑑2𝑦 𝑑𝑦
((A)) 𝑑𝑧 2 + 𝑑𝑧 + 𝑦 = 𝑒 𝑧
𝑑2𝑦
((B)) 𝑑𝑧 2 + 𝑦 = 𝑒 𝑧
𝑑2𝑦
((C)) 𝑑𝑧 2 + 𝑦 = 𝑒 𝑥
𝑑2𝑦
((D)) 𝑑𝑧 2 − 𝑦 = 𝑒 𝑧
((E)) B
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// On putting 𝑥 = 𝑒 𝑧 the transformed D.E. of 𝑥 2 −𝑥 + 4𝑦 = cos(log𝑥) + 𝑥sin(log𝑥) using 𝐷 ≡
𝑑𝑥 2 𝑑𝑥
𝑑
is
𝑑𝑧
((A)) (𝐷2 − 𝐷 + 4)𝑦 = sin𝑧 + 𝑒 𝑧 cos𝑧
((B)) (𝐷2 − 2𝐷 + 4)𝑦 = cos(log𝑥) + 𝑥sin(log𝑥)
((C)) (𝐷2 + 2𝐷 + 4)𝑦 = cos𝑧 + 𝑒 −𝑧 sin𝑧
((D)) (𝐷2 − 2𝐷 + 4)𝑦 = cos𝑧 + 𝑒 𝑧 sin𝑧
((E)) D
((F))

𝑑2𝑦 𝑑𝑦 𝑑
((Q))1_// On putting 𝑥 = 𝑒 𝑧 the transformed D.E. of 𝑥 2 2 − 3𝑥 + 5𝑦 = 𝑥 2 sin(log𝑥 ) using 𝐷 ≡ is
𝑑𝑥 𝑑𝑥 𝑑𝑧
((A)) (𝐷2 − 4𝐷 + 5)𝑦 = 𝑒 2𝑧 sin𝑧
((B)) (𝐷2 − 4𝐷 + 5)𝑦 = 𝑥 2 sin(log𝑥)
((C)) (𝐷2 − 4𝐷 − 4)𝑦 = 𝑒 𝑧 sin𝑧
2
((D)) (𝐷2 − 3𝐷 + 5)𝑦 = 𝑒 𝑧 sin𝑧
((E)) A
((F))

𝑑2𝑦 𝑑𝑦 𝑥3 𝑑
((Q))1_// On putting 𝑥 = 𝑒 𝑧 the transformed D.E. of 𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 − 𝑦 = 1+𝑥 2 using 𝐷 ≡ 𝑑𝑧 is
 𝑥3
((A)) (𝐷2 − 1)𝑦 = 1+𝑥 2
𝑒 3𝑧
((B)) (𝐷2 − 2𝐷 − 1)𝑦 = 1+𝑒 2𝑧
𝑒 3𝑧
((C)) (𝐷2 − 1)𝑦 = 1+𝑒 2𝑧
3
𝑒𝑧
((D)) (𝐷2 − 1)𝑦 = 2
1+𝑒 𝑧
((E)) C
((F))

𝑑2𝑦 𝑑𝑦 𝑑
((Q))1_// On putting 𝑥 = 𝑒 𝑧 the transformed D.E .of 𝑥 2 − 5𝑥 + 5𝑦 = 𝑥 2 log𝑥 using 𝐷 ≡ is
𝑑𝑥 2 𝑑𝑥 𝑑𝑧
𝑧2
((A)) (𝐷2 − 5𝐷 + 5)𝑦 = 𝑧 𝑒
((B)) (𝐷2 − 5𝐷 − 5)𝑦 = 𝑒 2𝑧 𝑧
((C)) (𝐷2 − 6𝐷 + 5)𝑦 = 𝑥 2 log𝑥
((D)) (𝐷2 − 6𝐷 + 5)𝑦 = 𝑧 𝑒 2𝑧
((E)) D
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// On putting 2𝑥 + 1 = 𝑒 𝑧 the transformed D.E. of (2𝑥 + 1)2 𝑑𝑥 2 − 2(2𝑥 + 1) 𝑑𝑥 + 5𝑦 = 6𝑥 using 𝐷 ≡
𝑑
is
𝑑𝑧
3
((A)) (𝐷2 − 2𝐷 − 3)𝑦 = 4 (𝑒 𝑧 − 1)
((B)) (𝐷2 + 2𝐷 + 3)𝑦 = 3(𝑒 𝑧 − 1)
3
((C)) (𝐷2 + 2𝐷 − 12)𝑦 = 4 (𝑒 𝑧 − 1)
((D)) (𝐷2 − 2𝐷 − 3)𝑦 = 6𝑥
((E)) A
((F))

𝑑2𝑦 𝑑𝑦
((Q))1_// On putting 3𝑥 + 2 = 𝑒 𝑧 the transformed D.E. of (3𝑥 + 2)2 𝑑𝑥 2 + 3(3𝑥 + 2) 𝑑𝑥 − 36𝑦 = 3𝑥 2 + 4𝑥 +
𝑑
1 using 𝐷 ≡ 𝑑𝑧 is
1
((A)) (𝐷2 + 3𝐷 − 36)𝑦 = 27 (𝑒 2𝑧 − 1)
1
((B)) (𝐷2 + 4)𝑦 = 9 (𝑒 2𝑧 − 1)
1
((C)) (𝐷2 − 4)𝑦 = (𝑒 2𝑧 − 1)
27
((D)) (𝐷2 − 9)𝑦 = (𝑒 2𝑧 − 1)
((E)) C
((F))

𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE 𝑑𝑡
+ 2𝑥 − 3𝑦 = 𝑡, 𝑑𝑡
− 3𝑥 + 2𝑦 = 𝑒 2𝑡 solution of 𝑦 using 𝐷 ≡ 𝑑𝑡 is
obtain from
((A)) (𝐷2 + 4𝐷 − 5)𝑥 = 1 + 2𝑡 + 3𝑒 2𝑡
((B)) (𝐷2 − 4𝐷 − 5)𝑦 = 𝑡 − 4𝑒 2𝑡
((C)) (𝐷2 + 4𝐷 + 5)𝑦 = 3𝑡 + 2𝑒 2𝑡
((D)) (𝐷2 + 4𝐷 − 5)𝑦 = 3𝑡 + 4𝑒 2𝑡
((E)) D
((F))

𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE + 2𝑥 − 3𝑦 = 𝑡, − 3𝑥 + 2𝑦 = 𝑒 2𝑡 solution of 𝑥 using 𝐷 ≡ is
𝑑𝑡 𝑑𝑡 𝑑𝑡
obt((A))in from
((A)) (𝐷2 + 4𝐷 − 5)𝑥 = 1 + 2𝑡 + 3𝑒 2𝑡
((B)) (𝐷2 − 4𝐷 − 5)𝑥 = 1 + 2𝑡 − 3𝑒 2𝑡
((C)) (𝐷2 + 4𝐷 − 5)𝑥 = 3𝑡 + 3𝑒 2𝑡
((D)) (𝐷2 + 4𝐷 − 5)𝑦 = 3𝑡 + 4𝑒 2𝑡
((E)) A
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑
((Q))2_// For the simultaneous Linear DE − 3𝑥 − 6𝑦 = 𝑡 2 , + 𝑑𝑡 − 3𝑦 = 𝑒 𝑡 solution of 𝑥 using 𝐷 ≡ 𝑑𝑡 is
𝑑𝑡 𝑑𝑡
obt((A))in from
((A)) (𝐷2 + 9)𝑥 = 6𝑒 𝑡 − 3𝑡 2 + 2𝑡
((B)) (𝐷2 + 9)𝑦 = −2𝑒 𝑡 − 2𝑡
((C)) (𝐷2 − 9)𝑥 = 6𝑒 𝑡 − 3𝑡 2
((D)) (𝐷2 + 12𝐷 + 9)𝑥 = 6𝑒 𝑡 + 3𝑡 2 + 2𝑡
((E)) A
((F))

𝑑𝑢 𝑑𝑣 𝑑
((Q))2_// For the simultaneous Linear DE 𝑑𝑥 + 𝑣 = sin𝑥, + 𝑢 = cos𝑥 solution of 𝑢 using 𝐷 ≡ 𝑑𝑥 is obtain
𝑑𝑥
from
((A)) (𝐷2 + 1)𝑢 = 2cos𝑥
((B)) (𝐷2 − 1)𝑢 = 0
((C)) (𝐷2 − 1)𝑢 = sin𝑥 − cos𝑥
((D)) (𝐷2 − 1)𝑣 = −2sin𝑥
((E)) B
((F))

𝑑𝑢 𝑑𝑣 𝑑
((Q))2_// For the simultaneous Linear DE 𝑑𝑥 + 𝑣 = sin𝑥, + 𝑢 = cos𝑥 solution of 𝑣 using 𝐷 ≡ 𝑑𝑥 is obtain
𝑑𝑥
from
((A)) (𝐷2 + 1)𝑣 = 0
((B)) (𝐷2 − 1)𝑢 = 0
((C)) (𝐷2 − 1)𝑣 = −2sin𝑥
((D)) (𝐷2 + 1)𝑣 = sin𝑥 + cos𝑥
((E)) C
((F))

𝑑𝑥 𝑑𝑦
((Q))2_// For the simultaneous Linear DE 𝐿 𝑑𝑡 + 𝑅𝑥 + 𝑅(𝑥 − 𝑦) = 𝐸, 𝐿 + 𝑅𝑦 − 𝑅(𝑥 − 𝑦) = 0 where
𝑑𝑡
𝑑
𝐿 , 𝑅 and 𝐸are constants, solution of 𝑥 using 𝐷 ≡ 𝑑𝑡 is obtain from
((A)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 5𝑅2 )𝑥 = 2𝑅𝐸 + 2𝑅
((B)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 3𝑅2 )𝑦 = 𝑅𝐸
((C)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 3𝑅2 )𝑥 = 2𝑅𝐸
((D)) (𝐿2 𝐷2 + 2𝑅𝐿𝐷 + 5𝑅2 )𝑥 = 2𝑅𝐸
((E)) C
((F))

𝑑𝑥 𝑑𝑦
((Q))2_// For the simultaneous Linear DE 𝐿 𝑑𝑡 + 𝑅𝑥 + 𝑅(𝑥 − 𝑦) = 𝐸, 𝐿 + 𝑅𝑦 − 𝑅(𝑥 − 𝑦) = 0 where
𝑑𝑡
𝑑
𝐿 , 𝑅 and 𝐸 are constants, solution of 𝑦 using 𝐷 ≡ 𝑑𝑡 is obtain from
((A)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 5𝑅2 )𝑦 = 𝑅𝐸 + 2𝑅
((B)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 3𝑅2 )𝑦 = 𝑅𝐸
((C)) (𝐿2 𝐷2 + 4𝑅𝐿𝐷 + 3𝑅2 )𝑥 = 2𝑅𝐸
((D)) (𝐿2 𝐷2 + 2𝑅𝐿𝐷 + 5𝑅2 )𝑦 = 2𝑅𝐸
((E)) B
((F))

𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE 𝑑𝑡 + 𝑦 = 𝑒 𝑡 , + 𝑥 = 𝑒 −𝑡 solution of 𝑥 using 𝐷 ≡ 𝑑𝑡 is obtain from
𝑑𝑡

((A)) (𝐷2 − 1)𝑥 = 2𝑒 𝑡

((B)) (𝐷2 − 1)𝑦 = −𝑒 𝑡 − 𝑒 −𝑡
((C)) (𝐷2 + 1)𝑥 = 𝑒 −𝑡 + 𝑒 𝑡
((D)) (𝐷2 − 1)𝑥 = 𝑒 𝑡 − 𝑒 −𝑡
((E)) D
((F))

𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE + 5𝑥 − 2𝑦 = 𝑡, + 2𝑥 + 𝑦 = 0, solution of 𝑦 using 𝐷 ≡ 𝑑𝑡 is
𝑑𝑡 𝑑𝑡
obtain from
((A)) (𝐷2 − 6𝐷 − 9)𝑦 = 2𝑡
((B)) (𝐷2 + 6𝐷 + 9)𝑥 = 1 + 𝑡
((C)) (𝐷2 + 6𝐷 + 1)𝑦 = 𝑡
((D)) (𝐷2 + 6𝐷 + 9)𝑦 = 2𝑡
((E)) B
((F))

𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE 𝑑𝑡 + 𝑦 = 𝑒 𝑡 , + 𝑥 = 𝑒 −𝑡 , solution of 𝑦 using 𝐷 ≡ 𝑑𝑡 is obtain from
𝑑𝑡
((A)) (𝐷2 − 1)𝑦 = 2𝑒 𝑡
((B)) (𝐷2 − 1)𝑦 = −𝑒 𝑡 − 𝑒 −𝑡
((C)) (𝐷2 + 1)𝑦 = 𝑒 −𝑡 + 𝑒 𝑡
((D)) (𝐷2 − 1)𝑥 = 𝑒 𝑡 − 𝑒 −𝑡
((E)) B
((F))
 𝑑𝑥 𝑑𝑦 𝑑
((Q))2_// For the simultaneous Linear DE + 5𝑥 − 2𝑦 = 𝑡, + 2𝑥 + 𝑦 = 0, solution of 𝑥 using 𝐷 ≡ 𝑑𝑡 is
𝑑𝑡 𝑑𝑡
obtain from
((A)) (𝐷2 + 6𝐷 + 9)𝑥 = 1 + 𝑡
((B)) (𝐷2 − 6𝐷 + 9)𝑥 = 2𝑡
((C)) (𝐷2 + 6𝐷 + 1)𝑥 = 𝑡
((D)) (𝐷2 + 6𝐷 + 9)𝑦 = 2𝑡
((E)) A
((F))

𝑑𝑢 𝑑𝑣
((Q))1_// For the D.E. + 𝑣 = sin 𝑥, + 𝑢 = cos 𝑥, the auxiliary equation for 𝑢 is
𝑑𝑥 𝑑𝑥
((A)) 𝐷2 + 1 = 0
((B)) 𝐷2 + 2 = 0
((C)) 𝐷2 − 1 = 0
((D)) 𝐷2 − 2 = 0
((E)) C
((F))

((Q))1_// The general form of symmetric simultaneous DE is

−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((A)) 𝑎0 𝑑𝑥 𝑛 + 𝑎1 𝑑𝑥 𝑛−1 + 𝑎2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥), where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
𝑑𝑥 𝑑𝑦 𝑑𝑧
((B)) = = , where 𝑃, 𝑄, 𝑅 are function of 𝑥, 𝑦, 𝑧.
𝑃 𝑄 𝑅
𝑛 −1 −2
𝑛𝑑 𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((C)) 𝑎0 𝑥 + 𝑎1 𝑥 𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥),where 𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
𝑑𝑥 𝑛
−1 −2
𝑑𝑛𝑦 𝑑𝑛 𝑦 𝑑𝑛 𝑦
((D)) 𝑎0 (𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 𝑛 + 𝑎1 (𝑎𝑥 + 𝑏)𝑛−1 𝑑𝑥 𝑛−1 + 𝑎2 (𝑎𝑥 + 𝑏)𝑛−2 𝑑𝑥 𝑛−2 + ⋯ ⋯ + 𝑎𝑛 𝑦 = 𝑓(𝑥),where
𝑎0 , 𝑎1 , 𝑎2 ⋯ ⋯ , 𝑎𝑛 are constant
((E)) B
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Considering the first two ratio of the symmetrical simultaneous DE = −𝑥𝑦 = 𝑥(𝑧−2𝑦), one of the
𝑦2
relation in the solution of DE is
((A)) 𝑥 2 + 𝑦 2 = 𝑐
((B)) 𝑥 3 + 𝑦 3 = 𝑐
𝑥2 𝑦3
((C)) − 2 = 3 + 𝑐
((D)) 𝑥 2 − 𝑦 2 = 𝑐
((E)) A
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Which of the following are the sets of multipliers for the solution of = 4𝑥−2𝑧 = 2𝑦−3𝑥
3𝑥−4𝑦
((A)) 𝑥, 𝑦, 𝑧
((B)) 4, 3, 2
1 1 1
((C)) 𝑥 , 𝑦 , 𝑧
1 1 1
((D)) 4 , 3 , 2
((E)) A
((F))

𝑥𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Considering the first & third ratio of the symmetrical simultaneous DE = 𝑥 2𝑧 = 𝑦 3, one of the
𝑦3𝑧
relation in the solution of DE is
((A)) 𝑥 2 − 𝑧 2 = 𝑐
((B)) 𝑥 4 − 𝑦 4 = 𝑐
((C)) 𝑥 3 − 𝑧 3 = 𝑐
((D)) 𝑥 − 𝑧 = 𝑐
((E)) A
((F))

𝑥𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Using a set of multiplier as 1, 𝑦, 𝑧 the solution of DE 𝑧 2−2𝑦𝑧−𝑦2 = 𝑦+𝑧 = 𝑦−𝑧 is
((A)) 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑐
𝑦2 𝑧2
((B)) 𝑥 + 2 + 2 = 𝑐
((C)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((D)) 𝑥 + 𝑦 2 + 𝑧 2 = 𝑐
((E)) A
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Considering the second & third ratio of the symmetrical simultaneous DE = 2𝑥𝑦 = 2𝑥𝑧, one
𝑥 2 −𝑦 2 −𝑧 2
of the relation in the solution of DE is
1 1
((A)) 𝑦 2 − 𝑧 2 = 𝑐
((B)) 𝑦 2 − 𝑧 2 = 𝑐
((C)) 𝑦 = 𝑐 𝑧
((D)) 𝑥 − 𝑧 = 𝑐
((E)) C
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Using a set of multiplier as 1,1,1 the solution of DE 𝑦−𝑧 = 𝑧−𝑥 = 𝑥−𝑦 is
((A)) 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑐
((B)) 𝑥 − 𝑦 − 𝑧 = 𝑐
((C)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((D)) −𝑥 + 𝑦 − 𝑧 = 𝑐
((E)) C
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Using a set of multiplier as 𝑥 3 , 𝑦 3 , 𝑧 3 the solution of DE 𝑥(2𝑦 4−𝑧 4) = 𝑦(𝑧 4−2𝑥 4) = 𝑧(𝑥 4−𝑦 4) is
((A)) 𝑥 3 + 𝑦 3 + 𝑧 3 = 𝑐
((B)) 𝑥 4 + 𝑦 4 + 𝑧 4 = 𝑐
((C)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((D)) 𝑥 𝑦𝑧 = 𝑐
((E)) B
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Considering the first two ratio of the symmetrical simultaneous DE = = , one of the
𝑦2𝑧 𝑥 2𝑧 𝑦2𝑥
relation in the solution of DE is
((A)) 𝑥 2 − 𝑦 2 = 𝑐
((B)) 𝑥 − 𝑦 = 𝑐
((C)) 𝑥 3 − 𝑦 3 = 𝑐
((D)) 𝑥 3 + 𝑦 3 = 𝑐
((E)) C
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Using a set of multiplier as 3, 2, 1 the solution of DE = −𝑥 = 2𝑥−3𝑦 is
𝑦
((A)) 3𝑥 2 + 2𝑦 2 + 𝑧 2 = 𝑐
3 2 1
((B)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((C)) 3𝑥 − 2𝑦 − 𝑧 = 𝑐
((D)) 3𝑥 + 2𝑦 + 𝑧 = 𝑐
((E)) D
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Using a set of multiplier as 𝑥 , 𝑦, 𝑧 the solution of DE = = is
3𝑧−4𝑦 4𝑥−2𝑧 2𝑦−3𝑥
3 3 3
((A)) 𝑥 + 𝑦 + 𝑧 = 𝑐
1 1 1
((B)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((C)) 𝑥 + 𝑦 + 𝑧 = 𝑐
((D)) 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑐
((E)) D
((F))

𝑑𝑥 𝑑𝑦 𝑑𝑧
((Q))1_// Considering the first two ratio of the symmetrical simultaneous DE = = 𝑥 2𝑦 2𝑧 2, one of the
𝑦2 𝑥2
relation in the solution of DE is
1 1
((A)) 𝑥 − 𝑦 = 𝑐
((B)) 𝑥 − 𝑦 = 𝑐
((C)) 𝑥 2 − 𝑦 2 = 𝑐
((D)) 𝑥 3 − 𝑦 3 = 𝑐
((E)) D
((F))