Find differential equation by eliminating a and b from y = ex (a cos x + b sin x).

Question 36 of 50

Q. Find differential equation by eliminating a and b from y = e^x (a cos x + b sin x).

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2022Subjective· 5mImportance★★★★★

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Differentiate twice, then algebraically eliminate the two arbitrary constants $a,b$ to leave a relation purely in $y,y',y''$ .

Given $y = e^x(a\cos x+b\sin x)$ .

Differentiate once (product rule):

$y' = e^x(a\cos x+b\sin x) + e^x(-a\sin x+b\cos x)$

$= e^x\big[(a+b)\cos x+(b-a)\sin x\big]$

Differentiate again:

$y'' = e^x\big[(a+b)\cos x+(b-a)\sin x\big] + e^x\big[-(a+b)\sin x+(b-a)\cos x\big]$

$= e^x\big[(a+b+b-a)\cos x + (b-a-a-b)\sin x\big]$

$= e^x\big[2b\cos x - 2a\sin x\big]$

Eliminate $a,b$ . Form $y''-2y'+2y$ and check the coefficients of $\cos x$ and $\sin x$ cancel:

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Q.Find differential equation by eliminating a and b from y = e^x (a cos x + b sin x).