Question. The order of the differential equation of all tangent lines to the parabola y = x^{2} is

(a) 1

(b) 2

(c) 3

(d) 4

Answer:

(a) 1

Question.The differential equation of all parabolas whose axis of symmetry is along the axis of the x-axis is of order

(a) 3

(b) 1

(c) 2

(d) none of these

Answer:

(c) 2

Question.The degree of the equation satisfying the relation \(\sqrt{1+x^{2}}+\sqrt{1+y^{2}}=\lambda(\sqrt{1+y^{2}}-y \sqrt{1+x^{2}})\) is

(a) 1

(b) 2

(c) 3

(d) none of these

Answer:

(a) 1

Question. The degree of the differential equation \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2 / 3}+4-\frac{3 d y}{d x}=0\) is

(a) 2

(b) 1

(c) 3

(d) none of these

Answer:

(a) 2

Question. The differential equation satisfied by y = \(\frac{A}{x}\) + B is (A, B are parameters)

(a) x^{2} y_{1} = y

(b) xy_{1} + 2y_{2} = 0

(c) xy_{2} + 2y_{1} = 0

(d) none of these

Answer:

(c) xy_{2} + 2y_{1} = 0

Question. The differential equation having solution is y = 17e^{x} + ae^{-x} is

(a) y” – x = 0

(b) y” – y = 0

(c) y’ – y = 0

(d) y’ – x = 0

Answer:

(b) y” – y = 0

Question. The differential equation representing the family of ellipses with centre at origin and foci on x-axis is given as

(a) xy’ + y = 0

(b) x^{2}y^{2}(y”)^{2} + yy’= 0

(c) xyy” + x(y’)^{2} – yy’ = 0

(d) None of these

Answer:

(b) x^{2}y^{2}(y”)^{2} + yy’= 0

Question. The differential equation of all parabolas whose axes are along x-axis is

(a) \(y_{2}^{2}+y_{1}=0\)

(b) \(y_{1}^{2}+y_{2}=0\)

(c) \(y_{1}^{2}+y_{1} y_{2}=0\)

(d) \(y_{1}^{2}+y y_{2}=0\)

Question. The equation of family of curves for which the length of the normal is equal to the radius vector is

(a) \(y^{2} \mp x^{2}=k^{2}\)

(b) \(y \pm x=k\)

(c) y^{2} = kx

(d) none of these

Question. The solution of the differential equation \(\frac{d y}{d x}=\frac{x^{2}+y^{2}+1}{2 x y}\) satisfying (1) = 1, is

(a) a hyperbola

(b) a circle

(c) y^{2} = x(1 + x) – 10

(d) (x – 2)^{2} + (y – 3)^{2} = 5xy

Question. Given the differential equation \(\frac{d y}{d x}=\frac{6 x^{2}}{2 y+\cos y}\); y(1) = π

Mark out the correct statement.

(a) solution is y^{2} – sin y = -2x^{3} + C

(b) solution is y^{2} + sin y = 2x^{3} + C

(c) C = π^{2}+ 2√2

(d) C = π^{2} + 2

Question. For the differential equation \(x \frac{d y}{d x}+2 y=x y \frac{d y}{d x}\),

(a) order is 1 and degree is 1

(b) solutio is ln(yx^{2}) = C – y

(c) order is 1 and degree is 2

(d) solution is ln(xy^{2}) = C + y

Question. The particular solution In(\(\frac{d y}{d x}\)) = 3x + 4y, y(0) = 0 is

(a) e^{3x} + 3e^{-4y} = 4

(b) 4e^{3x} – 3e^{-4y} = 3

(c) 3e^{3x} + 4e^{4y} = 7

(d) 4e^{3x} + 3e^{-4y} = 7

Question. If ydx – xdy + ln x dx = 0, y(1) = -1, then

(a) y + 1 + ln x = 0

(b) y + 1 + 2 ln x = 0

(c) 2(y + 1) + lnx = 0

(d) y + 1 – y ln x = 0

Question. The differential equation \(\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{y}}\) determines a family of circle with

(a) variable radii and fixed centre (0, 1)

(b) variable radii and fixed centre (0, -1)

(c) fixed radius 1 and variable centre on x-axis

(d) fixed radius 1 and variable centre on y-axis

Question. If y dx + y^{2} dy = x dy, x ∈ R, y > 0 and y(1) = 1, then y(-3) =

(a) 3

(b) 2

(c) 1

(d) 5

Question. If (x + y)^{2} \(\frac{d y}{d x}\) = a^{2}, y = 0 when x = 0, then y = a if \(\frac{x}{a}\) =

(a) 1

(b) tan 1

(c) tan 1 + 1

(d) tan 1 – 1

Question. If sinx \(\frac{d y}{d x}\) + y cosx = x sinx, then (y – 1) sinx =

(a) c – x sinx

(b) c + xcosx

(c) c – x cos x

(d) c + x sin x

Question. The solution of differential equation (e^{y} + 1) cosx dx + e^{y} sinx dy = 0 is

(a) (e^{y} + 1) sinx = c

(b) e^{x} sinx = c

(c) (e^{x} + 1) cosx = c

(d) none of these

Question. The general solution of the differential equation \(\frac{d y}{d x}=\frac{x^{2}}{y^{2}}\) is

(a) x^{3} – y^{3} = c

(b) x^{3} + y^{3} = c

(c) x^{2} + y^{2} = c

(d) x^{2} – y^{2} = c

Question. The solution of differential equation \(\frac{d y}{d x}=\frac{x-y}{x+y}\) is

(a) x^{2} – y^{2} + 2xy + c = 0

(b) x^{2} – y^{2} – xy + c = 0

(c) x^{2} – y^{2} + xy + c = 0

(d) x^{2} – y^{2} – 2xy + c = 0

Question. The order and degree of the differential equation \(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x^{\frac{1}{5}}=0\) respectively are

(a) 2 and not defined

(b) 2 and 2

(c) 2 and 3

(d) 3 and 3

Question. Integrating factor of the differential equation \(\frac{d y}{d x}\) + y tanx – sec x = 0 is

(a) cos x

(b) sec x

(c) e^{cos x}

(d) e^{sec x}

Question. The solution of the differential equation \(\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}\) is

(a) y = tan^{-1} x

(b) y – x = k(1 + xy)

(c) x = tan^{-1} y

(d) tan(xy) = k

Question. The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is

(a) \(\frac{\sin x}{\sin y}=c\)

(b) sin x sin y = c

(c) sin x + sin y = c

(d) cos x cos y = c