Question

Which of the following differential equations has y = c₁e^x + c₂e^–x as the general solution?
$fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y space equals space 0$
$fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y space equals space 0$
$fraction numerator straight d squared straight y over denominator dx squared end fraction plus 1 space equals space 0$
$fraction numerator straight d squared straight y over denominator dx squared end fraction minus 1 space equals space 0$

Solution

fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y space equals space 0

The given equation is
$straight y space equals straight c subscript 1 straight e to the power of straight x plus straight c subscript 2 straight e to the power of negative straight x end exponent$ ...(1)
$therefore space space space space space dy over dx space equals space straight c subscript 1 straight e to the power of straight x minus straight c subscript 2 straight e to the power of negative straight x end exponent rightwards double arrow space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight c subscript 1 straight e to the power of straight x plus straight c subscript 2 straight e to the power of negative straight x end exponent space space space rightwards double arrow space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight y space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y space equals space 0 comma space space which space is space the space required space differential space equation.$
$therefore$ (B) is the correct answer.

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Vector Algebra

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

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Vector Algebra

Which of the following differential equations has y = c1ex + c2e–x as the general solution?