Find the general solution of each of the following equation:
$left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space 2 space straight x cubed space straight y space where space straight y apostrophe space equals space dy over dx$

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Solution

The given differential equation is

left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space space 2 straight x cubed straight y

open parentheses straight x minus 1 close parentheses space dy over dx space equals space 2 straight x cubed straight y space space space space space space space space space space space space space or space space space space space dy over dx space equals space fraction numerator 2 straight x cubed straight y over denominator straight x minus 1 end fraction.

Separating the variables, we get,

right enclose table row cell 1 over straight y dy space equals space fraction numerator 2 straight x cubed over denominator straight x minus 1 end fraction dx end cell row cell or space space 1 over straight y dy space equals space open parentheses 2 straight x squared plus 2 straight x plus 2 plus fraction numerator 2 over denominator straight x minus 1 end fraction close parentheses dx end cell row cell table row cell Integrating comma space we space get comma end cell row cell log space open vertical bar straight y close vertical bar space equals space 2 straight x cubed over 3 plus fraction numerator 2 straight x squared over denominator 2 end fraction plus 2 straight x end cell row cell table row cell plus 2 space log space open vertical bar straight x minus 1 close vertical bar plus straight A end cell row cell or space log space open vertical bar straight y close vertical bar minus 2 space log space open vertical bar straight x minus 1 close vertical bar end cell row cell equals 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A end cell end table end cell end table end cell end table end enclose space table row cell table row cell table row cell table row cell stack straight x minus 1 long division enclose space 2 straight x cubed end enclose with 2 straight x squared plus 2 straight x plus 2 on top end cell row cell space space space space space space 2 straight x cubed minus 2 straight x squared end cell row cell negative space space plus space end cell end table end cell row _________ row cell 2 space straight x squared end cell end table end cell row cell 2 straight x squared minus 2 straight x end cell row cell negative space space space plus end cell end table end cell row ___________ row cell table row cell 2 space straight x end cell row cell 2 straight x minus 2 end cell row cell table row cell negative space plus end cell row __________ row 2 end table end cell end table end cell end table 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

log space open vertical bar fraction numerator straight y over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction close vertical bar space equals 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A space space space space space space rightwards double arrow space space space open vertical bar fraction numerator straight y over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction close vertical bar space equals space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A end exponent

straight y space equals straight e to the power of straight x space left parenthesis straight x minus 1 right parenthesis squared space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x end exponent space space space space or space space space space straight y space equals straight e space left parenthesis straight x minus 1 right parenthesis squared space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x end exponent

where e is an arbitrary constant.

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Vector Algebra

Find the general solution of each of the following equation:
$left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space 2 space straight x cubed space straight y space where space straight y apostrophe space equals space dy over dx$

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Vector Algebra

Find the general solution of each of the following equation:

Some More Questions From Vector Algebra Chapter

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

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the general solution of each of the following equation:
$left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space 2 space straight x cubed space straight y space where space straight y apostrophe space equals space dy over dx$

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