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QUESTION: 1

The degree of the differential equation satisfying

Solution:

putting x = sin A and y = sin B in the given relation, we get

cos ,4 + cos B = a(sin A - sin B)

⇒ A B = 2 cot^{-1} a

⇒ sin^{-1} x - sin^{-1} y = 2 cot^{-1} a Differentiating w.r.t. x, we get

Clearly, it is a differential equation of degree one

QUESTION: 2

The differential equation whose solution is Ax^{2} + By^{2} = 1, where A and B are arbitrary constants, is of

Solution:

QUESTION: 3

Differential equation of the family of circles touching the line y = 2 at (0, 2) is

Solution:

QUESTION: 4

The differential equation of all parabolas whose axis are parallel to the y-axis is

Solution:

The equation of a member of the family of parabolas having axis parallel toy-ax is is

y = Ax^{2} + Bx + C ......(1)

where A, B, and C are arbitrary constants

Differentiating equation (1) w.r.t. x, we .....(2)

which on again differentiating w.r.t. jc gives ......(3)

Differentiating (3) w.r.t. x, we get

QUESTION: 5

The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

Solution:

) be the centre on y-axis then its radius will be k as it passes through origin. Hence its equation is

QUESTION: 6

The differential equation whose general solution is given by, y = where c_{1}, c_{2}, c_{3}, c_{4}, c_{5} are arbitrary constants,

is

Solution:

QUESTION: 7

The equation of the curves through the point (1, 0) and whose slope is

Solution:

QUESTION: 8

The solution of the equation log(dy/dx) = ax + by is

Solution:

QUESTION: 9

Solution of differential equation dy – sin x sin ydx = 0 is

Solution:

QUESTION: 10

The solution of the equation

Solution:

Putting u = x - y, we get du/dx = 1 - dy/dx. The given equation can be written as 1 - du/dx = cos u

QUESTION: 11

Solution:

QUESTION: 12

The general solution of the differential equation

Solution:

QUESTION: 13

The solutions of (x + y + 1) dy = dx is

Solution:

Putting x + y 1 = u, we have du = dx + dy and the given equations reduces to u(du - dx) = dx

QUESTION: 14

The slope of the tangent at (x, y) to a curve passing through is given by then the equation of the curve is

Solution:

On integration, we get

This passes through (1,π/4), therefore 1 = log C.

QUESTION: 15

The solution of (x^{2} + xy)dy = (x^{2} + y^{2})dx is

Solution:

∴ equation reduces to

QUESTION: 16

The solution of (y + x + 5)dy = (y – x + 1) dx is

Solution:

The intersection of y – x + 1 = 0 and y + x + 5 = 0 is (-2, -3).

Put x = X - 2, y = Y - 3

The given equation reduces to

putting Y = vX, we get

QUESTION: 17

The slope of the tangent at (x, y) to a curve passing through a point (2, 1) is then the equation of the curve is

Solution:

∴ equation (1) transforms to

QUESTION: 18

The solution of satisfying y(1) = 1 is given by

Solution:

Rewriting the given equation as

Hence y^{2} = x(1 +x) - 1 which represents a system of hyperbola.

QUESTION: 19

Solution of the equation where

Solution:

The given differential equation can be written as which is linear differential equation of first order.

QUESTION: 20

A function y = f(x) satisfies (x + 1) f ' (x) - 2 (x^{2} + x)f(x) = 5, then f(x) is

Solution:

QUESTION: 21

The general solution of the equation

Solution:

QUESTION: 22

The solution of the differential equation

Solution:

Integrating, we get the solutions

QUESTION: 23

Which of the following is not the differential equation of family of curves whose tangent from an angle of π/4 with the hyperbola xy = c^{2}?

Solution:

QUESTION: 24

Tangent to a curve intercepts the y-axis at a point P. A line perpendicular to this tangent through P passes through another point (1, 0). The differential equation of the curve is

Solution:

The equations of the tangent at the point

The coor dinates of the point P are

The slope of the perpendiculer line

which is the rquried differential equation to the curve at y = f(x).

QUESTION: 25

The curve for which the normal at any point (x, y) and the line joining the origin to that point from an isosceles triangle with the x-axis as base is

Solution:

It is given that the triangle OPC is an isosceles triangle.

Therefore, OM= MG = sub-normal

On integration, we get x^{2} -y^{2} = C, which is a rectangular hyperbola

QUESTION: 26

The curve satisfying the equation and passing through the point (4, - 2 ) is

Solution:

It passes through the point (4, -2).

QUESTION: 27

A normal at P(x, y) on a curve meets the x-axis at Q and N is the foot of the ordinate at then the equation of curve given that it passes through the point (3,1) is

Solution:

Let the equation of the curve be y = f(x).

It is lincar differential equation.

QUESTION: 28

The equation of the curve passing through (2, 7/2) and having gradient is

Solution:

This passes through (2, 7/2),

Thus the equation of the curve is

QUESTION: 29

A normal at any point (x, y) to the curve y = f(x) cuts a triangle of unit area with the axis, the differential equation of the curve is

Solution:

Equation of normal at point p is

Y – y = (dx/dy)(X – x)

QUESTION: 30

The differential equation of all parabola each of which has a latus rectum 4a and whose axis parallel to the x-axis is

Solution:

Equations to the family of parabolas is (y - k)^{2} = 4a (x - h)

(substing y - k from equations (1))

Hence the order is 2 and the degree is 1.

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