The value of α is_________ , where α ∈ R ,s.t the differential equation , y" + 2(α-1)(α-3)y + (α-2)y = 0,has a solution y(x) = a cos(αx) + b sin(βx) for some non-zero real nos a,b.Correct answer is '3'. Can you explain this answer?

Question

The value of &alpha; is_________ , where &alpha; &isin; R ,s.t the differential equation , y" + 2(&alpha;-1)(&alpha;-3)y + (&alpha;-2)y = 0,has a solution y(x) = a cos(&alpha;x) + b sin(&beta;x) for some non-zero real nos a,b.Correct answer is '3'. Can you explain this answer?

Answer

Explanation:

Given Differential Equation:
The given differential equation is y'' + 2(α-1)(α-3)y + (α-2)y = 0.

Assumed Solution:
Let the solution be y(x) = a cos(αx) + b sin(βx) for some non-zero real numbers a, b.

Substitute the Assumed Solution:
Substitute the assumed solution into the differential equation and simplify the terms.

Identify the Coefficients:
Compare the coefficients of cos(αx) and sin(βx) terms to find the values of α and β.

Obtain the Value of α:
By equating the coefficients of cos(αx) and sin(βx) terms, we get α = 3 as the value that satisfies the given conditions.

Conclusion:
Therefore, the value of α is 3, which satisfies the given differential equation and allows for a solution in the form y(x) = a cos(3x) + b sin(βx) for some non-zero real numbers a, b.

Answer

here given eqⁿ is,

here the solⁿ of this eqn is given by we know that differential equation will have such solution only when AE has complex root, whose real part is zero.

thus in equation (1), (α-1)(α-3) = 0 => a = 1,3

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