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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

(x³ + 2 y²) dx + 2xy dy = 0

The equation y²dx + (x² – xy – y²) dy = 0. which can be written as

dy over dx equals negative fraction numerator straight y squared over denominator straight x squared minus xy minus straight y squared end fraction space equals space minus fraction numerator open parentheses begin display style straight y over straight x end style close parentheses squared over denominator 1 minus open parentheses begin display style straight y over straight x end style close parentheses minus open parentheses begin display style straight y over straight x end style close parentheses squared end fraction comma space is space homogeneous space as space straight R. straight H. straight S. space is space of space form space straight F open parentheses straight y over straight x close parentheses

as R.H.S. is of form

straight F open parentheses straight y over straight x close parentheses

therefore space space space left parenthesis straight D right parenthesis space is space correct space answer.

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Short Answer Type

237.

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

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