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The general solution of the differential equation Dny + a1Dn-1y + a2Dn-2y +  ... + any = X of nth order,.where X is a function of x alone and a1, a2..., an are constants, is given by y = Complementary Function (C.F.’) + Particular Integral (P.I.) Then
  • a)
    C.F. involves no arbitrary constant but the P.I. involves n arbitrary' constants
  • b)
    C.F. involves n arbitrary constants but P.T. may or may not involve any arbitrary constant
  • c)
    C.F. involves n arbitrary constants but P.I. does not involve any
  • d)
    C.F. and P.I. both involve atleast one arbitrary constant.
Correct answer is option 'C'. Can you explain this answer?
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The general solution of the differential equation Dny +a1Dn-1y + a2Dn-...
Discussion : The given differential equation is Dny + a1Dn-1y + a2Dn-2y +  ... + any = X(x)...(i)
Corresponding homogeneous differential equation is
Dny + a1Dn-1y + a2Dn-2y +  ... + any = 0 ...(ii)
then the complementary function is defined as the general solution of the corresponding homogeneous differential equation (ii) and therefore it will invlove n arbitrary constants. The Particular solution is a solution of the complete equation (i) and does not involve any arbitrary constant.
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The general solution of the differential equation Dny +a1Dn-1y + a2Dn-...
Discussion : The given differential equation is Dny + a1Dn-1y + a2Dn-2y +  ... + any = X(x)...(i)
Corresponding homogeneous differential equation is
Dny + a1Dn-1y + a2Dn-2y +  ... + any = 0 ...(ii)
then the complementary function is defined as the general solution of the corresponding homogeneous differential equation (ii) and therefore it will invlove n arbitrary constants. The Particular solution is a solution of the complete equation (i) and does not involve any arbitrary constant.
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The general solution of the differential equation Dny +a1Dn-1y + a2Dn-...
) + Particular Integral (P.I.).

The Complementary Function (C.F.) is the general solution of the homogeneous equation, which is obtained by setting X = 0 in the given differential equation.

The Particular Integral (P.I.) is a specific solution of the non-homogeneous equation, which is obtained by finding a particular solution that satisfies the given X.

Therefore, the general solution of the given nth order differential equation is expressed as:
y = C.F. + P.I.
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The general solution of the differential equation Dny +a1Dn-1y + a2Dn-2y + ... + any = X of nth order,.where X is a function of x alone and a1, a2..., anare constants, is given by y = Complementary Function (C.F.’) + Particular Integral (P.I.) Thena)C.F. involves no arbitrary constant but the P.I. involves n arbitrary constantsb)C.F. involves n arbitrary constants but P.T. may or may not involve any arbitrary constantc)C.F. involves n arbitrary constants but P.I. does not involve anyd)C.F. and P.I. both involve atleast one arbitrary constant.Correct answer is option 'C'. Can you explain this answer?
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The general solution of the differential equation Dny +a1Dn-1y + a2Dn-2y + ... + any = X of nth order,.where X is a function of x alone and a1, a2..., anare constants, is given by y = Complementary Function (C.F.’) + Particular Integral (P.I.) Thena)C.F. involves no arbitrary constant but the P.I. involves n arbitrary constantsb)C.F. involves n arbitrary constants but P.T. may or may not involve any arbitrary constantc)C.F. involves n arbitrary constants but P.I. does not involve anyd)C.F. and P.I. both involve atleast one arbitrary constant.Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The general solution of the differential equation Dny +a1Dn-1y + a2Dn-2y + ... + any = X of nth order,.where X is a function of x alone and a1, a2..., anare constants, is given by y = Complementary Function (C.F.’) + Particular Integral (P.I.) Thena)C.F. involves no arbitrary constant but the P.I. involves n arbitrary constantsb)C.F. involves n arbitrary constants but P.T. may or may not involve any arbitrary constantc)C.F. involves n arbitrary constants but P.I. does not involve anyd)C.F. and P.I. both involve atleast one arbitrary constant.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The general solution of the differential equation Dny +a1Dn-1y + a2Dn-2y + ... + any = X of nth order,.where X is a function of x alone and a1, a2..., anare constants, is given by y = Complementary Function (C.F.’) + Particular Integral (P.I.) Thena)C.F. involves no arbitrary constant but the P.I. involves n arbitrary constantsb)C.F. involves n arbitrary constants but P.T. may or may not involve any arbitrary constantc)C.F. involves n arbitrary constants but P.I. does not involve anyd)C.F. and P.I. both involve atleast one arbitrary constant.Correct answer is option 'C'. Can you explain this answer?.
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