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Differential Equations

Short Answer Type

141.

Solve:
$open parentheses straight e to the power of straight x plus 1 close parentheses space straight y space dy space plus space left parenthesis straight y plus 1 right parenthesis space straight e to the power of straight x space dx space equals space 0$

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142.

Solve:
$straight y left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space 0.$

83 Views

143.

Solve:
$straight y space logydx space minus space straight x space dy space equals space 0$

91 Views

Long Answer Type

144.
Show that the general solution of the differential equation $dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals 0$ is given by (x + y + 1) = A (1 – x – y – 2 x y), where A is parameter.

The given differential equation is
$dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals space 0 space or space space fraction numerator dy over denominator straight y squared plus straight y plus 1 end fraction plus fraction numerator dx over denominator straight x squared plus straight x plus 1 end fraction space equals space 1$
Integrating, we get
$integral fraction numerator dy over denominator straight y squared plus straight y plus 1 end fraction plus integral fraction numerator dx over denominator straight x squared plus straight x plus 1 end fraction space equals space straight C$
$rightwards double arrow space space space space space integral space fraction numerator dy over denominator open parentheses straight y plus begin display style 1 half end style close parentheses squared plus open parentheses 1 minus begin display style 1 fourth end style close parentheses end fraction plus integral fraction numerator dx over denominator open parentheses straight x plus begin display style 1 half end style close parentheses squared plus 1 minus begin display style 1 fourth end style end fraction space equals space straight C rightwards double arrow space space space space integral space fraction numerator dy over denominator open parentheses straight y plus begin display style 1 half end style close parentheses squared plus open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style close parentheses squared end fraction plus integral fraction numerator dx over denominator open parentheses straight x plus begin display style 1 half end style close parentheses squared plus open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style close parentheses squared end fraction space equals space straight C rightwards double arrow space space space space fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight y plus begin display style 1 half end style over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction close parentheses plus fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus begin display style 1 half end style over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction close parentheses space equals space straight C rightwards double arrow space space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction close parentheses space plus space tan to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction close parentheses space equals space fraction numerator square root of 3 straight C over denominator 2 end fraction space equals space straight A subscript 1 space left parenthesis say right parenthesis$
Taking tangents on the two sides, we get
$fraction numerator begin display style fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction end style plus begin display style fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction end style over denominator 1 minus open parentheses begin display style fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction end style close parentheses space open parentheses begin display style fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction end style close parentheses end fraction space equals space tan space straight A subscript 1 space space rightwards double arrow space space space fraction numerator 2 square root of 3 space left parenthesis straight x plus straight y plus 1 right parenthesis over denominator 3 minus left parenthesis 4 space straight x space straight y space plus 2 straight x plus 2 straight y plus 1 right parenthesis end fraction equals space tan space straight A subscript 1$
$rightwards double arrow space space space space space space fraction numerator 2 square root of 3 left parenthesis straight x plus straight y plus 1 right parenthesis over denominator 2 left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis end fraction space equals space tan space straight A subscript 1 rightwards double arrow space space space space straight x plus straight y plus 1 space equals space fraction numerator 1 over denominator square root of 3 end fraction tan space straight A subscript 1 space left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis rightwards double arrow space space space straight x plus straight y plus 1 space equals space straight A left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis comma$
where $straight A space equals space fraction numerator 1 over denominator square root of 3 end fraction space space tan space straight A subscript 1 space is space an space arbitrary space constant. space$

77 Views

Short Answer Type

145.

Solve:
$dy over dx space equals space fraction numerator straight x left parenthesis 2 space log space straight x space plus space 1 right parenthesis over denominator sin space straight y space plus space straight y space cosy end fraction$

80 Views

146.

Solve
$dy over dx space equals space fraction numerator xe to the power of straight x logx plus straight e to the power of straight x over denominator straight x space cosy end fraction$

83 Views

Long Answer Type

147.

Solve
$dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x$

82 Views

148.

Solve:
$dy over dx space equals space cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root$

84 Views

149.

Solve:
$dy over dx equals negative straight x space sin squared straight x space space equals space fraction numerator 1 over denominator straight x space log space straight x end fraction$