Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Differential Equations

Multiple Choice Questions

541.

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

$(x^{2} + y^{2}) \frac{dy}{dx} - 2 xy = 0$
$(x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$
$(x^{2} - y^{2}) \frac{dy}{dx} - 2 xy = 0$
$(x^{2} + y^{2}) \frac{dy}{dx} + 2 xy = 0$

542.

The degree and order of the differential equation ${[1 + {(\frac{dy}{dx})}^{3}]}^{\frac{7}{3}} = 7 (\frac{d^{2} y}{{dx}^{2}})$ respectively are

3 and 7
3 and 2
7 and 3
2 and 3

543.

The particular solution of the differential equation

$y (1 + \log (x)) \frac{d x}{d y} - xlog (x) = 0$ , when, x = e, y = e² is

y = exlog(x)
ey = xlog(x)
xy = elog(x)
ylog(x) = ex

544.

If $\sin (x)$ is the integrating factor (IF) of the linear differential equation $\frac{dy}{dx} + Py = Q$ , then P is

$\log (\sin (x))$
$\cos (x)$
$\tan (x)$
$\cot (x)$

545.

The solution of the differential equation

$\frac{dy}{dx} = \tan (\frac{y}{x}) + \frac{y}{x}$ is

$\cos (\frac{y}{x}) = cx$
$\sin (\frac{y}{x}) = cx$
$\cos (\frac{y}{x}) = cy$
$\sin (\frac{y}{x}) = cy$

546.
The differential equation of all parabolas whose axis is Y-axis, is

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

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

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

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

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

$Axis of parabola = Y axis and vertex of parabola is (0, k) . ∴ Equation of parabola is {(x - 0)}^{2} = 4 a (y - k) x^{2} = 4 ay - 4 ak On differentiate both sides w . r . t,' x', we get 2 x = 4 a \frac{dy}{dx} x = 2 a \frac{dy}{dx} ∴ \frac{1}{2 a} = \frac{1}{x} \frac{dy}{dx} On differentiating both sides w . r . t' x', we get \frac{d}{dx} (\frac{1}{x} . \frac{dy}{dx}) = \frac{d}{dx} (\frac{1}{2 a}) \Rightarrow \frac{1}{x} . \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (- \frac{1}{x^{2}}) = 0 ∴ x \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} = 0$

547.

The particular solution of the differential equation xdy + 2ydx = 0, when x = 2, y = 1 is

xy = 4
x²y = 4
xy² = 4
x²y² = 4

548.

The general solution of the equation $\frac{dy}{dx} = \frac{y^{2} - x}{2 y (x + 1)}$ is

y² = (1 + x)log(1 + x) - c
$y^{2} = (1 + x) \log (\frac{c}{(1 - x)}) - 1$
$y^{2} = (1 - x) \log (\frac{c}{(1 - x)}) - 1$
$y^{2} = (1 + x) \log (\frac{c}{(1 + x)}) - 1$

549.

The general solution of the differential equation $\frac{dy}{dx} + \sin (\frac{x + y}{2}) = \sin (\frac{x - y}{2})$ is

$\log_{e} |\tan (\frac{y}{2})| = - 2 \sin (\frac{x}{2}) + C$
$\log_{e} |\tan (\frac{y}{4})| = 2 \sin (\frac{x}{2}) + C$
$\log_{e} (\tan (\frac{y}{2})) = - 2 \sin (\frac{x}{2}) + C$
$\log_{e} (\tan (\frac{y}{4})) = - 2 \sin (\frac{x}{2}) + C$

550.

The function y specified implicitly by the relation $\int_{0}^{y} e^{t} dt + \int_{0}^{x} \cos (t) dt$ = 0 satisfies the differential equation

$e^{2 y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$
$e^{y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (2 x)$
$e^{y} (2 \frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$
$e^{y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

546. The differential equation of all parabolas whose axis is Y-axis, is xd2ydx2 - dydx = 0 xd2ydx2 + dydx = 0 d2ydx2 - y = 0 d2ydx2 - dydx = 0

546.
The differential equation of all parabolas whose axis is Y-axis, is

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

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

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

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