Differential Equations अवकल समीकरण Chapter 9

By | January 29, 2021

Question. The order of the differential equation of all tangent lines to the parabola y = x2 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) x2 y1 = y
(b) xy1 + 2y2 = 0
(c) xy2 + 2y1 = 0
(d) none of these
Answer:
(c) xy2 + 2y1 = 0

 

Question. The differential equation having solution is y = 17ex + 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) x2y2(y”)2 + yy’= 0
(c) xyy” + x(y’)2 – yy’ = 0
(d) None of these
Answer:
(b) x2y2(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) y2 = 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) y2 = 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 y2 – sin y = -2x3 + C
(b) solution is y2 + sin y = 2x3 + C
(c) C = π2+ 2√2
(d) C = π2 + 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(yx2) = C – y
(c) order is 1 and degree is 2
(d) solution is ln(xy2) = C + y

 

Question. The particular solution In(\(\frac{d y}{d x}\)) = 3x + 4y, y(0) = 0 is
(a) e3x + 3e-4y = 4
(b) 4e3x – 3e-4y = 3
(c) 3e3x + 4e4y = 7
(d) 4e3x + 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 + y2 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 \(\frac{d y}{d x}\) = a2, 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 (ey + 1) cosx dx + ey sinx dy = 0 is
(a) (ey + 1) sinx = c
(b) ex sinx = c
(c) (ex + 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) x3 – y3 = c
(b) x3 + y3 = c
(c) x2 + y2 = c
(d) x2 – y2 = c

 

Question. The solution of differential equation \(\frac{d y}{d x}=\frac{x-y}{x+y}\) is
(a) x2 – y2 + 2xy + c = 0
(b) x2 – y2 – xy + c = 0
(c) x2 – y2 + xy + c = 0
(d) x2 – y2 – 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) ecos x
(d) esec 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

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