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Differential Equations

Question
CBSEENMA12033172
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Find one-parameter families of solution curves of the following differential equation:
 y' - y = cos 2x

Solution
The given differential equation is
                    straight y apostrophe minus straight y space equals space cos space 2 straight x space space space space space space space space space space space space space space space space space space space space space space space space space space or space space space dy over dx minus straight y space equals space cos space 2 straight x
Comparing it with dy over dx plus straight P space straight y space equals straight Q comma space we space get comma space space straight P equals space minus 1 comma space space straight Q space equals space cos space 2 straight x
                            integral straight P space dx space equals space minus integral 1 space dx space equals space minus straight x comma space space space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x
therefore space space space solution space of space differential space equation space is
                 straight y. space straight e to the power of straight x space equals space integral cos space 2 straight x. space straight e to the power of negative straight x end exponent space dx plus straight c space space or space space space straight y space straight e to the power of negative straight x end exponent space equals space integral straight e to the power of negative straight x end exponent space cos space 2 straight x space dx plus straight c
or        straight y space straight e to the power of straight x space equals space fraction numerator 1 over denominator square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared end root end fraction straight e to the power of negative straight x end exponent space cos space open square brackets 2 straight x minus tan to the power of negative 1 end exponent open parentheses fraction numerator 2 over denominator space space space space space space space space space minus 1 end fraction close parentheses close square brackets plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space 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 space integral straight e to the power of ax cos space bx space dx space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction straight e to the power of ax space cos space open parentheses bx minus tan to the power of negative 1 end exponent straight b over straight a close parentheses plus straight c close square brackets space space space space space space or space space space space space straight y space straight e to the power of negative straight x end exponent space equals space fraction numerator 1 over denominator square root of 5 end fraction straight e to the power of negative straight x end exponent space cos space left square bracket 2 straight x space minus space tan to the power of negative 1 end exponent left parenthesis negative 2 right parenthesis right square bracket space plus space straight c or space space space space space space space straight y space equals space fraction numerator 1 over denominator square root of 5 end fraction cos space left square bracket 2 straight x space minus tan to the power of negative 1 end exponent left parenthesis negative 2 right parenthesis right square bracket space plus space straight c space straight e to the power of straight x

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