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Differential Equations

Multiple Choice Questions

11.

If f''(0) = k, $k \neq 0$ , then the value of $\lim_{x \to 0} \frac{2 f (x) - 3 f (2 x) + f (4 x)}{x^{2}}$ is

12.

If y = $e^{{msin}^{- 1} (x)}$ then $(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - x \frac{d y}{d x} - ky = 0$ , where k is equal to

m²
2
- 1
- m²

13.
Solution of ${(x + y)}^{2} \frac{d y}{d x} = a^{2}$ ( 'a' being a constant) is

$\frac{(x + y)}{a} = \tan (\frac{y + C}{a})$ , C is an arbitrary

$xy = atan (Cx)$ , C is an arbitrary

$\frac{x}{a} = \tan (\frac{y}{C})$ , C is an arbitrary

$xy = \tan (x + C)$ , C is an arbitrary

$\frac{(x + y)}{a} = \tan (\frac{y + C}{a})$ , C is an arbitrary

We have,

${(x + y)}^{2} \frac{d y}{d x} = a^{2}$

Let x + y = v

$\Rightarrow 1 + \frac{d y}{d x} = \frac{d v}{d x}$

$\Rightarrow \frac{d y}{d x} = \frac{d v}{d x} - 1$

$∴ v^{2} (\frac{d v}{d x} - 1) = a^{2}$

$\Rightarrow v^{2} \frac{d v}{d x} = v^{2} + a^{2}$

$\Rightarrow \frac{d v}{d x} = \frac{v^{2} + a^{2}}{v^{2}}$

$\Rightarrow \frac{v^{2}}{v^{2} + a^{2}} dv = dx$

On integrating both sides, we get

$\int \frac{v^{2}}{v^{2} + a^{2}} dv = \int dx$

$\Rightarrow \int (1 - \frac{a^{2}}{v^{2} + a^{2}}) dv = x + C'$

$\Rightarrow v - \frac{a^{2}}{a} \tan^{- 1} (\frac{v}{a}) = x + C'$

$\Rightarrow x + y - {atan}^{- 1} (\frac{x + y}{a}) = x + C'$

$\Rightarrow y = {atan}^{- 1} (\frac{x + y}{a}) + C'$

$\Rightarrow \frac{y - C'}{a} = \tan^{- 1} (\frac{x + y}{a})$

$\Rightarrow (\frac{x + y}{a}) = \tan (\frac{y + C}{a})$

14.

The integrating factor of the first order differential equation

$x^{2} (x^{2} - 1) \frac{d y}{d x} + x (x^{2} + 1) y = x^{2} - 1$ is

e^x
$x - \frac{1}{x}$
$x + \frac{1}{x}$
$\frac{1}{x^{2}}$

15.

Let a and B be the roots of x² + x + 1 = 0. If n be a positive integer, then $α^{n} + β^{n}$ is

$2 \cos (\frac{2 nπ}{3})$
$2 \sin (\frac{2 nπ}{3})$
$2 \cos (\frac{nπ}{3})$
$2 \sin (\frac{nπ}{3})$

16.

For real x, the greatest value of $\frac{x^{2} + 2 x + 4}{2 x^{2} + 4 x + 9}$ is

1
- 1
$\frac{1}{2}$
$\frac{1}{4}$

17.

If the solution of the differential equation $x \frac{d y}{d x} + y = {xe}^{x} be xy = e^{x} ϕ (x) + C$ , then $ϕ (x)$ is equal to

x + 1
x - 1
1 - x
x

18.

The order of the differential equation of all parabolas whose axis of symmetry along X-axis is

2
3
1
None of these

19.

General solution of $y \frac{d y}{d x} + {by}^{2} = acos (x), 0 < x < 1$ is

$y^{2} = 2 a (2 bsin (x) + \cos (x)) + {ce}^{- 2 bx}$
$(4 b^{2} + 1) y^{2} = 2 a (\sin (x) + 2 bcos (x)) + {ce}^{- 2 bx}$
$(4 b^{2} + 1) y^{2} = 2 a (\sin (x) + 2 bcos (x)) + {ce}^{2 bx}$
$y^{2} = 2 a (2 bsin (x) + \cos (x)) + {ce}^{- 2 bx}$

20.

The integrating factor of the differential equation

$(1 + x^{2}) \frac{d y}{d x} + y = e^{\tan^{- 1} (x)}$ is

$\tan^{- 1} (x)$
1 + x²
$e^{\tan^{- 1} (x)}$
$\log_{e} (1 + x^{2})$

Previous Year Papers

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Test Yourself

Multiple Choice Questions

13. Solution of x + y2dydx = a2 ( 'a' being a constant) is x + ya = tany + Ca, C is an arbitrary xy = atanCx, C is an arbitrary xa = tanyC, C is an arbitrary xy = tan(x + C), C is an arbitrary

13.
Solution of ${(x + y)}^{2} \frac{d y}{d x} = a^{2}$ ( 'a' being a constant) is

$\frac{(x + y)}{a} = \tan (\frac{y + C}{a})$ , C is an arbitrary

$xy = atan (Cx)$ , C is an arbitrary

$\frac{x}{a} = \tan (\frac{y}{C})$ , C is an arbitrary

$xy = \tan (x + C)$ , C is an arbitrary