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QUESTION: 1

Let ( y - C)^{2} = Cx be the primitive of The no. of integral curves which will pass through (1, 2) is,

Solution:

QUESTION: 2

Which of the following transformations reduce the differential equation into the form

Solution:

Given differential equation is

Here which shows that the transformation t= 1/log z

reduce the given differential equation into the form

QUESTION: 3

has a solution ___________, when t= 0, then

Solution:

then its solution is

Now,

QUESTION: 4

The solution of the differential equation ^{}

where y(0) = 0 and y'(0) = -2 is

Solution:

Auxiliary equation is m^{2} - m-2 = 0

m=-1,2

∴ C.F = c_{1}e^{-x }+c_{2}e^{2x}

Solving these, we get c_{1}= 1, c_{2}= -1

∴ the required solution is y = e^{-x} - e^{2x}+ xe^{2x}

QUESTION: 5

An integrating factor of the differential equation

Solution:

**Correct Answer :- d**

**Explanation : **

QUESTION: 6

The differential equation representing the family of circles touching x-axis at the origin is

Solution:

The equation of the family of the circles touching x-axis at the origin is

x^{2} + (y-a)^{2} = a^{2}

QUESTION: 7

A curve passing through (2,3) and satisfying the differential equation

Solution:

differential w.r. to x

QUESTION: 8

The differential equation

(3a^{2}x^{2} + by cos x)dx + (2 sin x - 4ay^{3})dy = 0 is exact for

Solution:

Given differential equation is

(3a^{2}x^{2} + by cos x)dx + (2 sin x - 4ay^{3})dy = 0

since it si exact if

Here M = 3a^{2}x^{2} + bycosx

N = 2sinx - 4ay^{3}

then bcosx = 2cosx

=> cosx(b - 2) = 0

=> b= 2[∵ cos x ≠ 0]

which shows that given differential equation is exact for any value of a but b = 2

QUESTION: 9

The integrating factor of the equation (1 + y^{2})dx = (tan^{-1}y - x)dy is -

Solution:

QUESTION: 10

Solution of the differential equation _{}

Solution:

The given equation can be re-written as

Integrating

Putting y^{2} = v so that 2y (dy/dx) = (dv/dx). We then get

which is linear. Its I.F =

QUESTION: 11

Solution of equation , given y(0) = 1/2 is

Solution:

x=0, y=1/2 (1) ⇒ -2 = -3 +c ⇒ c=1

QUESTION: 12

Integrating factor of differential equation _{}

Solution:

**Correct Answer :- b**

**Explanation : **xcosx dy/dx + y(xsinx + cosx) = 1......(1)

=> dy/dx + y(tanx + 1/x) = secx/x............(2)

I.F of (2) e∫^{(tanx + 1/x) dx}

= e^{ln secx + lnx}

= e^{ln secx}

= x secx

I.F of (1) is xsecx/xcosx

= sec^{2} x

QUESTION: 13

the solution of differential equation _{}

Solution:

Hence the given differential equation

QUESTION: 14

The differential equation

Solution:

It is clear by the definition of linear differential equation and order of differential equation, that is linear and of second order.

QUESTION: 15

If (c_{1} logx + c_{2})x^{2} is the general solution of the differential equation then k equals

Solution:

Given differential equation is

--(1)

and it’s general solution is

(c_{1} log x + c_{2}) x^{2} ..(2)

Since (1) is homogeneous differential equation so put then (1) reduce to

It’s A.E is m^{2} + m(k - 1) + 4 = 0 • • (3)

Now from (2) we have

By observation we can say that (3) has solution m = 2,

... (4)

On comparing (3) and (4). we have

k-1 = -4

k=-3

*Multiple options can be correct

QUESTION: 16

Consider a function g which has derivative g’(x) for every real x and which satisfies g’(0) = 2 and g(x + y) = e^{y} g(x) + e^{x} g(y) for all x and y. Then which of the followings is/are correct ?

Solution:

*Multiple options can be correct

QUESTION: 17

Consider the differential equation (3x^{2}y^{4} + 2xy)dx+(2x^{3}y^{3} - x^{2})dy=0 then which of the follwing(s) is/are not an I.F

Solution:

Given differential equation is

*Multiple options can be correct

QUESTION: 18

If y_{1} and y_{2} are two solution of the differential equation . Then which of the followings can be given as general soln of this differential eqn.

Solution:

Option (a), (b) are clearly correct

Now we have

Now from (4) and (5)

*Multiple options can be correct

QUESTION: 19

For the given differential equation, which of the following(s) statement are true,

Solution:

Given equation ca be written as,

not homogenous differential equation

compare with

Here at x = 0, P and Q both are not defined.

⇒ x=0 is a singular point of given differential equation.

Now consider x P(x) = -1/2

and both are defined and differential at x = 0

⇒ x= 0 is a regular singular point of given equation.

*Multiple options can be correct

QUESTION: 20

Given that φ(x) = x^{2} is a solution of the differential equation, then which of the following(s) is/are not its 2^{nd } L.l. solution.

Solution:

we have x^{2} y" - 2y = 0

*Answer can only contain numeric values

QUESTION: 21

If the integrating factor of the differential equation, (x^{7}y^{2 }+ 3y)dx + 3x^{8}y - x) dy = 0 is of the form x^{m}y^{n} , then the sum of value of m and n is __________.

Solution:

Given differential equation is

(x^{7}y^{2 }+ 3y)dx + 3x^{8}y - x) dy = 0 ..(1)

If Integrating factor is x^{m}y^{n} then multiplying the given differential equation by I.F. , equation (1) becomes exact

i.e

is exact differential equation.

and for exact

on solving (1) and (2) we get

m=-7, n=1

m+n = -7+1 = -6

*Answer can only contain numeric values

QUESTION: 22

The sum of order and degree of the differential equation representing the family o f parabolas whose center is (-a.O) i s _______ .

Solution:

The equation of the families of parabolas whose centre (-a,0) is

y^{2} = 4a(x+a) ..(1)

Differentiating both sides with respect to x, we get

Dividing (i) by (ii) we get

Substituting this value of a in (2), a is eliminated and we get

which shows that order of this differential equation is one and degree is 2. The sum of order of this differential equation and degree is 3.

*Answer can only contain numeric values

QUESTION: 23

The value o f y as t -> ∞ , for an initial value o f y(1) = 0, for the differential equation

Solution:

Hence solution of equation (i) is,

But, y(1)=0, therefore

From equation (ii), we have

From equation (ii), we have

*Answer can only contain numeric values

QUESTION: 24

If y = f(x) be a particular solution of differential equation y" + 4y = 4 cosec^{2} 2x , then f(x) at is _________.

Solution:

*Answer can only contain numeric values

QUESTION: 25

Consider the differential equation __ __ be its general solution and u(x) = x^{3} then the value of v(x) at x=(-1) is, _________

Solution:

where

v(x)= -1/6x^{2}

∴ v(x)= -1/6(-1)^{2} = 0.1666666

*Answer can only contain numeric values

QUESTION: 26

No. of regular Singular points of the differential equation, x^{2}( x - 2 )^{2} y" + 2x (x - 2) y’+ (x + 1)y = 0. is __________.

Solution:

In the normalized form (6,3), this is

Clearly the singular points of the differential equation are x = 0 and x = 2. We investigate them one at a time.

Consider x = 0 first. and form the functions defined by the products

of the form. The product function defined by x^{2}P_{2}(x)f is analytical at x = 0, but that defined by xP_{1}(x) is not. Thus x = 0 is an irregular singular point of Now consider x = 2. Forming the products for this point, we have

Both of the product functions thus defined are analytic at x = 2. and hence x = 2 is a regular singular point.

*Answer can only contain numeric values

QUESTION: 27

The value of I.F of the differential equation, at x = 2 is, _______.

Solution:

*Answer can only contain numeric values

QUESTION: 28

The directional derivative of f(x,y) = x^{2} + xy at P_{o}(1,2) in the direction of the unit vector

Solution:

The derivative of f at P_{0}(x_{0},y_{o}) in the direction of the unit vector u = u_{1i} + u_{2}j is the number

*Answer can only contain numeric values

QUESTION: 29

Find the directional Derivative of f(x,y) = x^{2} sin 2y at the point

Solution:

*Answer can only contain numeric values

QUESTION: 30

Find the derivation of f(x,y,z) = x^{3} - xy^{2} -z at P_{o} (1,1,0) in the direction of v= 2i - 3j+ 6k

Solution:

the partial derivatives of f at P_{0} are

The gradient of f ata p_{0} is

The derivative of f at P_{0 }in the direction of v is therefore

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