If general solution of the differential equation ay" + by" + cy’ + dy = 0 is linearly spanned by ex, sinx and cosx, then which one of the following holds?a)a + b - c - d = 0b)a + b + c + d = 0c)a - b + c - d = 0d)a - b - c - d=0Correct answer is option 'B'. Can you explain this answer?

Question

If general solution of the differential equation ay" + by" + cy&rsquo; + dy = 0 is linearly spanned by ex, sinx and cosx, then which one of the following holds?a)a + b - c - d = 0b)a + b + c + d = 0c)a - b + c - d = 0d)a - b - c - d=0Correct answer is option 'B'. Can you explain this answer?

Answer

Answer

The general solution of the differential equation ay'' + by' + cy = 0 can be written as y = C1e^(r1x) + C2e^(r2x), where r1 and r2 are the roots of the characteristic equation ar^2 + br + c = 0.

If the roots are real and distinct, the general solution is y = C1e^(r1x) + C2e^(r2x).

If the roots are real and equal, the general solution is y = (C1 + C2x)e^(rx).

If the roots are complex conjugates, the general solution is y = e^(px)(C1cos(qx) + C2sin(qx)), where p and q are the real and imaginary parts of the complex root.

These are the general forms of the solution for the given differential equation. The specific values of C1, C2, r1, r2, p, and q would depend on the specific values of a, b, and c in the differential equation.

Answer

A=1 b=0 c=0 d=-1

Answer

Yes

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