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If m1 and m2 are the roots of the auxiliary equation of the given linear differential equation of second order with constant coefficients, then
- a)m1 and m2 are always real and distinct
- b)m1 and m2 are always complex
- c)m1 and m2 both may be complex and equal
- d)m1 and m2 both may be real and equal real and distinct or complex
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If m1and m2 are the roots of the auxiliary equation of the given linea...
The auxiliary equation is given by
a0D2 + a1D + a2 = 0 .....(i)
Its roots are given by
Clearly,
a0D2 + a1D + a2 = 0 .....(i)
Its roots are given by
Clearly,
Most Upvoted Answer
If m1and m2 are the roots of the auxiliary equation of the given linea...
The given linear differential equation of second order with constant coefficients can be written as:
a(d^2y/dx^2) + b(dy/dx) + cy = 0
where a, b, and c are constants.
The auxiliary equation for this differential equation is obtained by assuming a solution of the form y = e^(mx), where m is a constant. Substituting this into the differential equation, we get:
a(m^2e^(mx)) + b(me^(mx)) + ce^(mx) = 0
Simplifying this equation gives us the auxiliary equation:
am^2 + bm + c = 0
This is a quadratic equation in m, and its roots m1 and m2 are the solutions to the auxiliary equation.
Now let's go through each option to determine which one is correct:
a) m1 and m2 are always real and distinct:
This is not always true. The roots of the auxiliary equation can be real and distinct, but they can also be complex or equal. So option a is incorrect.
b) m1 and m2 are always complex:
This is also not always true. The roots of the auxiliary equation can be real or complex. So option b is incorrect.
c) m1 and m2 both may be complex and equal:
This is true. The roots of the auxiliary equation can be complex and equal. For example, if the quadratic equation has a discriminant of zero, then the roots will be complex and equal. So option c is correct.
d) m1 and m2 both may be real and equal, real and distinct, or complex:
This is also true. The roots of the auxiliary equation can be real and equal, real and distinct, or complex. So option d is correct.
In conclusion, the correct answer is option d. The roots m1 and m2 of the auxiliary equation can be real and equal, real and distinct, or complex.
a(d^2y/dx^2) + b(dy/dx) + cy = 0
where a, b, and c are constants.
The auxiliary equation for this differential equation is obtained by assuming a solution of the form y = e^(mx), where m is a constant. Substituting this into the differential equation, we get:
a(m^2e^(mx)) + b(me^(mx)) + ce^(mx) = 0
Simplifying this equation gives us the auxiliary equation:
am^2 + bm + c = 0
This is a quadratic equation in m, and its roots m1 and m2 are the solutions to the auxiliary equation.
Now let's go through each option to determine which one is correct:
a) m1 and m2 are always real and distinct:
This is not always true. The roots of the auxiliary equation can be real and distinct, but they can also be complex or equal. So option a is incorrect.
b) m1 and m2 are always complex:
This is also not always true. The roots of the auxiliary equation can be real or complex. So option b is incorrect.
c) m1 and m2 both may be complex and equal:
This is true. The roots of the auxiliary equation can be complex and equal. For example, if the quadratic equation has a discriminant of zero, then the roots will be complex and equal. So option c is correct.
d) m1 and m2 both may be real and equal, real and distinct, or complex:
This is also true. The roots of the auxiliary equation can be real and equal, real and distinct, or complex. So option d is correct.
In conclusion, the correct answer is option d. The roots m1 and m2 of the auxiliary equation can be real and equal, real and distinct, or complex.
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Community Answer
If m1and m2 are the roots of the auxiliary equation of the given linea...
The auxiliary equation is given by
a0D2 + a1D + a2 = 0 .....(i)
Its roots are given by
Clearly,
a0D2 + a1D + a2 = 0 .....(i)
Its roots are given by
Clearly,
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If m1and m2 are the roots of the auxiliary equation of the given linear differential equation of second order with constant coefficients, thena)m1and m2 are always real and distinctb)m1 and m2 are always complexc)m1and m2 both may be complex and equald)m1and m2both may be real and equal real and distinct or complexCorrect answer is option 'D'. Can you explain this answer?
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If m1and m2 are the roots of the auxiliary equation of the given linear differential equation of second order with constant coefficients, thena)m1and m2 are always real and distinctb)m1 and m2 are always complexc)m1and m2 both may be complex and equald)m1and m2both may be real and equal real and distinct or complexCorrect answer is option 'D'. 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 If m1and m2 are the roots of the auxiliary equation of the given linear differential equation of second order with constant coefficients, thena)m1and m2 are always real and distinctb)m1 and m2 are always complexc)m1and m2 both may be complex and equald)m1and m2both may be real and equal real and distinct or complexCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If m1and m2 are the roots of the auxiliary equation of the given linear differential equation of second order with constant coefficients, thena)m1and m2 are always real and distinctb)m1 and m2 are always complexc)m1and m2 both may be complex and equald)m1and m2both may be real and equal real and distinct or complexCorrect answer is option 'D'. Can you explain this answer?.
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