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Differential Equations

Long Answer Type

231.

Solve the following differential equation:
$dy over dx equals fraction numerator straight x left parenthesis 2 straight y minus straight x right parenthesis over denominator straight x left parenthesis 2 straight y plus straight x right parenthesis end fraction comma space space if space space straight y space equals space 1 space space when space straight x space equals space 1.$

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232.

Solve the following initial value problem:
$dy over dx space equals space fraction numerator straight y left parenthesis straight x plus 2 straight y right parenthesis over denominator straight x left parenthesis 2 straight x plus straight y right parenthesis end fraction. space space space straight y left parenthesis 1 right parenthesis space equals space 2$

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Short Answer Type

233. Show that the family of curves for which the slope of the tangent at any point (x, y) on it is

fraction numerator straight x squared plus straight y squared over denominator 2 xy end fraction comma

is given by

straight x squared minus straight y squared space equals space straight c space straight x.

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Multiple Choice Questions

234. The general solution of the differential equation

fraction numerator straight y space dx space minus space straight x space dy over denominator straight y end fraction space equals space 0

xy = C
x = Cy²
y = Cx
y = Cx

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235.

A homogeneous differential equation of the from $dx over dy space equals space straight h open parentheses straight x over straight y close parentheses$ can be solved by making the substitution.

y = vx
v = yx
x = vy
x = vy

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236. Which of the following is a homogeneous differential equation?

(4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(xy) dx – (x³ + y³) dy = 0
(x³ + 2 y²) dx + 2xy dy = 0
(x³ + 2 y²) dx + 2xy dy = 0

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Short Answer Type

237.
Solve : $dy over dx plus straight y space equals space sin space straight x comma space space left parenthesis straight x space element of space straight R right parenthesis$

The given differential equation is $dy over dx plus straight y space equals space sin space straight x$
Comparing it with $dy over dx plus straight P space straight y space equals space straight Q comma space space space we space get comma space space space straight P space equals space 1 comma space space straight Q space equals space sin space straight x$
$therefore space space space integral straight P space dx space equals space integral 1 space dx space equals space straight x. space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x$
$therefore space space space$ solution of differential equation is
$straight y space. straight e to the power of straight x space equals space integral space sinx. space straight e to the power of straight x space dx space plus space straight c space or space space straight y space. straight e to the power of straight x space equals space fraction numerator 1 over denominator square root of 1 plus 1 end root end fraction straight e to the power of straight x space sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 over 1 close parentheses plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space integral straight e to the power of ax space sin space bx space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction straight e to the power of ax space sin space open parentheses bx minus tan to the power of negative 1 end exponent straight b over straight a close parentheses close square brackets or space space space straight y space equals space fraction numerator 1 over denominator square root of 2 end fraction sin space open parentheses straight x minus straight pi over 4 close parentheses plus space straight c space straight e to the power of negative straight x end exponent$

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Long Answer Type

238. Solve the differential equation:

straight x dy over dx minus straight y minus 2 straight x cubed space equals space 0.

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239. Solve the differential equation:

dy over dx minus 2 straight y space equals space 3 straight x.

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Short Answer Type

240.

Solve $dy over dx plus straight y over straight x space equals space straight e to the power of straight x comma space space left parenthesis straight x greater than 0 right parenthesis.$