Question

For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:
$straight y space equals straight a space straight e to the power of straight x space plus space straight b space straight e to the power of negative straight x end exponent plus straight x squared$ : $fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y plus straight x squared minus 2 space equals space 0$

Solution

Here, $straight y space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared$
$therefore space space space space space space dy over dx space equals space ae to the power of straight x minus be to the power of negative straight x end exponent plus 2 straight x therefore space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus 2 therefore space space space straight L. straight H. straight S. space equals space fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space equals straight a space straight e to the power of straight x plus be to the power of negative straight x end exponent plus 2 minus left parenthesis ae to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared right parenthesis plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus 2 minus ae to the power of straight x minus be to the power of negative straight x end exponent minus straight x squared plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space space space equals space 0 space space equals space straight R. straight H. straight S. therefore space space space space space straight y space equals space straight a space straight e to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared$
is a solution of the given differential equation.

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