The D.E whose solution is
$y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} sin x + c_{4} cos x$ is

y_{4} = y_{2}

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y_{4} = y_{3}

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y_{4} = y_{1}

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y_{4} = y

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Solution

The correct option is D $y_{4} = y$
$y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} sin x + c_{4} cos x$
$\Rightarrow y_{1} = c_{1} e^{x} - c_{2} e^{- x} + c_{3} cos x - c_{4} sin x$
$\Rightarrow$ $y_{2} = c_{1} e^{x} + c_{2} e^{- x} - c_{3} sin x - c_{4} cos x$
$\Rightarrow$ $y_{3} = c_{1} e^{x} - c_{2} e^{- x} - c_{3} cos x + c_{4} sin x$
$\Rightarrow$ $y_{4} = c_{1} e^{x} + c_{2} e^{- x} + c_{3} sin x + c_{4} cos x$
$\Rightarrow y_{4} = y$

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The D.E whose solution is y=c1ex+c2e−x+c3sinx+c4cosx is

The correct option is D y4=yy=c1ex+c2e−x+c3sinx+c4cosx⇒y1=c1ex−c2e−x+c3cosx−c4sinx⇒ y2=c1ex+c2e−x−c3sinx−c4cosx⇒y3=c1ex−c2e−x−c3cosx+c4sinx⇒y4=c1ex+c2e−x+c3sinx+c4cosx⇒y4=y

The D.E whose solution is
$y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} sin x + c_{4} cos x$ is