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Differential Equations

Multiple Choice Questions

11.

What are the order and degree, respectively, of the differential equation ${(\frac{d^{3} y}{{dx}^{3}})}^{2} = y^{4} + {(\frac{dy}{dx})}^{5} ?$

4, 5
2, 3
3, 2
5, 4

12.
The differential equation of the family of curves y = pcos (ax) + qsin(ax), where p, q are arbitrary constants, is

$\frac{d^{2} y}{{dx}^{2}} - a^{2} y = 0$

$\frac{d^{2} y}{{dx}^{2}} - ay = 0$

$\frac{d^{2} y}{{dx}^{2}} + ay = 0$

$\frac{d^{2} y}{{dx}^{2}} + a^{2} y = 0$

$\frac{d^{2} y}{{dx}^{2}} + a^{2} y = 0$

$y = pcos (ax) + qsin (ax) \Rightarrow \frac{dy}{dx} = - pasin (ax) + qacos (ax) \Rightarrow \frac{d^{2} y}{{dx}^{2}} = - {pa}^{2} \cos (ax) - {qa}^{2} \sin (ax) \Rightarrow \frac{d^{2} y}{{dx}^{2}} = - a^{2} [pcos (ax) + qsin (ax)] \Rightarrow \frac{d^{2} y}{{dx}^{2}} = - a^{2} u \Rightarrow \frac{d^{2} y}{{dx}^{2}} + a^{2} y = 0$

13.

$The equation of the curve passing through the point (- 1, - 2) which satisfies \frac{dy}{dx} = - x^{2} - \frac{1}{x^{3}}, is$

$17 x^{2} y - 6 x^{2} + 3 x^{5} - 2 = 0$
$6 x^{2} y + 17 x^{2} + 2 x^{5} - 3 = 0$
$6 xy - 2 x^{2} + 17 x^{5} + 3 = 0$
$17 x^{2} y + 6 xy - 3 x^{5} + 5 = 0$

14.

What is the order of the differential equation whose solution is

$y = acos (x) + bsin (x) + {ce}^{- x} = d, where a, b, c and d are arbitrary constants ?$

15.

$What is the solution of the differential equation \ln (\frac{dy}{dx}) = ax + by ?$

${ax}^{ax} + {be}^{by} = c$
$\frac{1}{a} e^{ax} + \frac{1}{b} e^{by} = 0$
${ae}^{ax} + {be}^{- bx} = c$
$\frac{1}{b} e^{ax} + \frac{1}{b} e^{- by} = c$

16.

$If μ = e^{ax} \sin (bx) and ν = e^{ax} \cos (bx), then what is μ \frac{du}{dx} + v \frac{dv}{dx} = ?$

ae^2ax
$(a^{2} + b^{2}) e^{ax}$
abe^2ax
$(a + b) e^{ax}$

17.

$If y = \sin (\ln (x)), then which one of the following is correct ?$

$\frac{d^{2} y}{{dx}^{2}} + y = 0$
$\frac{d^{2} y}{{dx}^{2}} = 0$
$x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y = 0$
$x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} + y = 0$

18.

$What is the solution of the differential equation \frac{dx}{dy} = \frac{x + y + 1}{x + y - 1} ?$

y - x + ln(x + y) = c
y + x + 2ln(x + y) = c
y - x + 2ln(x + y) = c
y + x + 2ln(x + y) = c

19.

$What is \lim_{x \to \frac{π}{6}} \frac{2 \sin^{2} (x) + \sin (x) - 1}{2 \sin^{2} (x) - 3 \sin (x) + 1} = ?$

$- \frac{1}{2}$
$- \frac{1}{3}$
- 2
- 3

20.

$What is \frac{d^{2} y}{{dy}^{2}} = ?$

$- {(\frac{d^{2} y}{{dx}^{2}})}^{- 1} {(\frac{dy}{dx})}^{- 3}$
${(\frac{d^{2} y}{{dx}^{2}})}^{- 1} {(\frac{dy}{dx})}^{- 2}$
$- (\frac{d^{2} y}{{dx}^{2}}) {(\frac{dy}{dx})}^{- 3}$
${(\frac{d^{2} y}{{dx}^{2}})}^{- 1}$

Previous Year Papers

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Multiple Choice Questions

12. The differential equation of the family of curves y = pcos (ax) + qsin(ax), where p, q are arbitrary constants, is d2ydx2 - a2y = 0 d2ydx2 - ay = 0 d2ydx2 + ay = 0 d2ydx2 + a2y = 0

12.
The differential equation of the family of curves y = pcos (ax) + qsin(ax), where p, q are arbitrary constants, is

$\frac{d^{2} y}{{dx}^{2}} - a^{2} y = 0$

$\frac{d^{2} y}{{dx}^{2}} - ay = 0$

$\frac{d^{2} y}{{dx}^{2}} + ay = 0$

$\frac{d^{2} y}{{dx}^{2}} + a^{2} y = 0$