Chapter 24

The differential equation of the family of curves, \(x^{2}=4 b(y+b), b \in R\), is: (a) \(x\left(y^{\prime}\right)^{2}=x+2 y y^{\prime}\) (b) \(x\left(y^{\prime}\right)^{2}=2 y y^{\prime}-x\) (c) \(x y^{\prime \prime}=y^{\prime}\) (d) \(x\left(y^{\prime}\right)^{2}=x-2 y y^{\prime}\)

The differential equation representing the family of ellipse having foci either on the \(x\)-axis or on the \(y\)-axis centre at the origin and passing through the point \((0,3)\) is: (a) \(x y y^{\prime}+y^{2}-9=0\) (b) \(x+y y^{\prime \prime}=0\) (c) \(x y y^{\prime \prime}+x\left(y^{\prime}\right)^{2}-y y^{\prime}=0\) (d) \(x y y^{\prime}-y^{2}+9=0\)

If the differential equation representing the family of all circles touching \(\mathrm{x}\)-axis at the origin is \(\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{g}(\mathrm{x}) \mathrm{y}\), then \(\mathrm{g}(\mathrm{x})\) equals: (a) \(\frac{1}{2} x\) (b) \(2 x^{2}\) (c) \(2 \mathrm{x}\) (d) \(\frac{1}{2} x^{2}\)

Statement-1: The slope of the tangent at any point P on a parabola, whose axis is the axis of \(x\) and vertex is at the origin, is inversely proportional to the ordinate of the point \(\mathrm{P}\) Statement-2: The system of parabolas \(y^{2}=4 a x\) satisfies a differential equation of degree 1 and order 1 . (a) Statement- 1 is true; Statement- 2 is true; Statement- 2 is a correct explanation for statement- 1 . (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.

Statement 1: The degrees of the differential equations \(\frac{d y}{d x}+y^{2}=x\) and \(\frac{d^{2} y}{d x^{2}}+y=\sin x\) are equal. Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined. (a) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 . (b) Statement 1 is false, Statement 2 is true. (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1 .

The differential equation which represents the family of curves \(y=c_{1} e^{c_{2} x}\), where \(c_{1}\), and \(c_{2}\) are arbitrary constants, is (a) \(y^{\prime \prime}=y^{\prime} y\) (b) \(y y^{\prime \prime}=y^{\prime}\) (c) \(y y^{\prime \prime}=\left(y^{\prime}\right)^{2}\) (d) \(y^{\prime}=y^{2}\)

The differential equation of all circles passing through the origin and having their centres on the x-axis is (a) \(y^{2}=x^{2}+2 x y \frac{d y}{d x}\) (b) \(y^{2}=x^{2}-2 x y \frac{d y}{d x}\) (c) \(x^{2}=y^{2}+x y \frac{d y}{d x}\) (d) \(x^{2}=y^{2}+3 x y \frac{d y}{d x}\)

The differential equation whose solution is \(A x^{2}+B y^{2}=1\) where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of (a) second order and second degree (b) first order and second degree (c) first order and first degree (d) second order and first degree

The differential equation representing the family of curves \(y^{2}=2 c(x+\sqrt{c})\), where \(c>0\), is a parameter, is of order and degree as follows: (a) order 1, degree 2 (b) order 1 , degree 1 (c) order 1, degree 3 (d) order 2, degree 2

The differential equation for the family of circle \(x^{2}+y^{2}-2 a y=0\), where a is an arbitrary constant is (a) \(\left(x^{2}+y^{2}\right) y^{\prime}=2 x y\) (b) \(2\left(x^{2}+y^{2}\right) y^{\prime}=x y\) (c) \(\left(x^{2}-y^{2}\right) y^{\prime}=2 x y\) (d) \(2\left(x^{2}-y^{2}\right) y^{\prime}=x y\)

The degree and order of the differential equation of the family of all parabolas whose axis is \(x\) - axis, are respectively. (a) 2,3 (b) 2,1 (c) 1,2 (d) 3,2 .

The order and degree of the differential equation \(\left(1+3 \frac{d y}{d x}\right)^{2 / 3}=4 \frac{d^{3} y}{d x^{3}}\) are \(\quad[\mathbf{2 0 0 2}]\) (a) \(\left(1, \frac{2}{3}\right)\) (b) \((3,1)\) (c) \((3,3)\) (d) \((1,2)\)

If \(y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosec} x\) is the solution of the differential equation, \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{p}(x) y=\frac{2}{\pi} \operatorname{cosec} x, 0

If \(y=y(x)\) is the solution of the differential equation \(\frac{5+\mathrm{e}^{x}}{2+y} \cdot \frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{e}^{x}=0\) satisfying \(y(0)=1\), then a value of \(y\left(\log _{e} 13\right)\) is: \(\quad\) [Sep. 05,2020 (I)] (a) 1 (b) \(-1\) (c) 0 (d) 2

The solution of the differential equation \(\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0\) is [Sep. 04, 2020 (II)] (where \(C\) is a constant of integration.) (a) \(x-\frac{1}{2}\left(\log _{e}(y+3 x)\right)^{2}=C\) (b) \(x-\log _{e}(y+3 x)=C\) (c) \(y+3 x-\frac{1}{2}\left(\log _{e} x\right)^{2}=C\) (d) \(x-2 \log _{e}(y+3 x)=C\)

Let \(f:(0, \infty) \rightarrow(0, \infty)\) be a differentiable function such that \(f(1)=e\) and \(\lim _{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}=0\). If \(f(x)=1\), then \(x\) is equal to: \(\quad\) [Sep. 04, 2020 (II)] (a) \(\frac{1}{e}\) (b) \(2 e\) (c) \(\frac{1}{2 e}\) (d) \(e\)

The solution curve of the differential equation, \(\left(1+e^{-x}\right)\left(1+y^{2}\right) \frac{d y}{d x}=y^{2}\), which passes through the point \((0,1)\), is: (a) \(y^{2}+1=y\left(\log _{e}\left(\frac{1+e^{-x}}{2}\right)+2\right)\) (b) \(y^{2}+1=y\left(\log _{e}\left(\frac{1+e^{x}}{2}\right)+2\right)\) (c) \(y^{2}=1+y \log _{e}\left(\frac{1+e^{x}}{2}\right)\) (d) \(y^{2}=1+y \log _{e}\left(\frac{1+e^{-x}}{2}\right)\)

If \(x^{3} d y+x y d x=x^{2} d y+2 y d x ; y(2)=e\) and \(x>1\), then \(y(4)\) is equal to : [Sep. 03, 2020 (II)] (a) \(\frac{3}{2}+\sqrt{e}\) (b) \(\frac{3}{2} \sqrt{e}\) (c) \(\frac{1}{2}+\sqrt{e}\) (d) \(\frac{\sqrt{e}}{2}\)

Let \(y=y(x)\) be the solution of the differential equation, \(\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}=-\cos x, y>0, y(0)=1 .\) If \(y(\pi)=a\) and \(\frac{d y}{d x}\) at \(x=\pi\) is \(b\), then the ordered pair \((a, b)\) is equal to : (a) \(\left(2, \frac{3}{2}\right)\) (b) \((1,-1)\) (c) \((1,1)\) (d) \((2,1)\)

If a curve \(y=f(x)\), passing through the point \((1,2)\), is the solution of the differential equation, \(2 x^{2} d y=\left(2 x y+y^{2}\right) d x\), then \(f\left(\frac{1}{2}\right)\) is equal to : [Sep. 02, 2020 (II)] (a) \(\frac{1}{1+\log _{e} 2}\) (b) \(\frac{1}{1-\log _{e} 2}\) (c) \(1+\log _{e} 2\) (d) \(\frac{-1}{1+\log _{e} 2}\)

If \(f 2(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2}

If \(\frac{d y}{d x}=\frac{x y}{x^{2}+y^{2}} ; y(1)=1\); then a value of \(x\) satisfying \(y(x)=e\) is: (a) \(\frac{1}{2} \sqrt{3} e\) (b) \(\frac{e}{\sqrt{2}}\) (c) \(\sqrt{2} e\) (d) \(\sqrt{3} e\)

Let \(f(x)=\left(\sin \left(\tan ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right)^{2}-1,|x|>1\). If \(\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1}(f(x))\right)\) and \(y(\sqrt{3})=\frac{\pi}{6}\), then \(y(-\sqrt{3})\) is equal to: (a) \(\frac{2 \pi}{3}\) (b) \(-\frac{\pi}{6}\) (c) \(\frac{5 \pi}{6}\) (d) \(\frac{\pi}{3}\)

Let \(y=y(x)\) be a solution of the differential equation, \(\sqrt{1-x^{2}} \frac{d y}{d x}+\sqrt{1-y^{2}}=0, \mid x k 1\) If \(y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}\), then \(y\left(\frac{-1}{\sqrt{2}}\right)\) is equal to: (a) \(\frac{\sqrt{3}}{2}\) (b) \(-\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{\sqrt{2}}\) (d) \(-\frac{\sqrt{3}}{2}\)

If \(y=y(x)\) is the solution of the differential equation, \(e^{y}=e^{x}\) such that \(y(0)=0\), then \(y(\mathrm{l})\) is equal to: (a) \(1+\log _{e} 2\) (b) \(2+\log _{e} 2\) (c) \(2 e\) (d) \(\log _{e} 2\)

The general solution of the differential equation \(\left(y^{2}-x^{3}\right)\) \(\mathrm{d} x-x y d y=0(x \neq 0)\) is: (a) \(y^{2}-2 x^{2}+c x^{3}=0\) (b) \(y^{2}+2 x^{3}+c x^{2}=0\) (c) \(y^{2}+2 x^{2}+c x^{3}=0\) (d) \(y^{2}-2 x^{3}+c x^{2}=0\) (where \(c\) is a constant of integration)

If \(\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0

Given that the slope of the tangent to a curve \(y=y(x)\) at any point \((x, y)\) is \(\frac{2 y}{x^{2}}\). If the curve passes through the centre of the circle \(x^{2}+y^{2}-2 x-2 y=0\), then its equation is: (a) \(x \log _{e}|y|=2(x-1)\) (b) \(x \log _{e}|y|=-2(x-1)\) (c) \(x^{2} \log _{e}|y|=-2(x-1)\) (d) \(x \log _{e}|y|=x-1\)

The solution of the differential equation, \(\frac{\mathrm{d} y}{\mathrm{~d} x}=(x-y)^{2}\), when \(y(1)=1\), is : (a) \(\log _{\mathrm{e}}\left|\frac{2-x}{2-y}\right|=x-y\) (b) \(-\log _{e}\left|\frac{\mid-x+y}{1+x-y}\right|=2(x-1)\) (c) \(-\log _{e}\left|\frac{1+x-y}{1-x+y}\right|=x+y-2\) (d) \(\log _{e}\left|\frac{2-y}{2-x}\right|=2(y-1)\)

If \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{3}{\cos ^{2} x} y=\frac{1}{\cos ^{2} x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)\), and \(y\left(\frac{\pi}{4}\right)=\frac{4}{3}\), then \(y\left(-\frac{\pi}{4}\right)\) equals: (a) \(\frac{1}{3}+\mathrm{e}^{6}\) (b) \(\frac{1}{3}\) (c) \(-\frac{4}{3}\) (d) \(\frac{1}{3}+\mathrm{e}^{3}\)

The curve amongst the family of curves represented by the differential equation, \(\left(x^{2}-y^{2}\right) d x+2 x y d y=0\) which passes through \((1,1)\), is: (a) a circle with centre on the \(x\)-axis. (b) an ellipse with major axis along the \(y\)-axis. (c) a circle with centre on the \(y\)-axis. (d) a hyperbola with transverse axis along the \(x\)-axis.

Let \(f:[0,1] \rightarrow R\) be such that \(f(x y)=f(x) \cdot f(y)\), for all \(x, y \in[0,1]\), and \(f(0) \neq 0\). If \(y=y(x)\) satisfies the differential equation, \(\frac{d y}{d x}=f(x)\) with \(y(0)=1\), then \(y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)\) equal to: (a) 3 (b) 4 (c) 2 (d) 5

If \((2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0\) and \(y(0)=1\), then \(y\left(\frac{\pi}{2}\right)\) is equal to : (a) \(\frac{4}{3}\) (b) \(\frac{1}{3}\) (c) \(-\frac{2}{3}\) (d) \(-\frac{1}{3}\)

If a curve \(y=f(x)\) passes through the point \((1,-1)\) and satisfies the differential equation, \(y(1+x y) d x=x d y\), then \(\mathrm{f}\left(-\frac{1}{2}\right)\) is equal to : (a) \(\frac{2}{5}\) (b) \(\frac{4}{5}\) (c) \(-\frac{2}{5}\) (d) \(-\frac{4}{5}\)

If \(f(x)\) is a differentiable function in the interval \(((0, \infty)\) such that \(\mathrm{f}(\mathrm{a})=1\) and \(\lim _{\mathrm{t} \rightarrow \mathrm{x}} \frac{\mathrm{t}^{2} \mathrm{f}(\mathrm{x})-\mathrm{x}^{2} \mathrm{f}(\mathrm{t})}{\mathrm{t}-\mathrm{x}}=1\), for each \(\mathrm{x}>0\). then \(\mathrm{f}\left(\frac{3}{2}\right)\) is equal to: (a) \(\frac{23}{18}\) (b) \(\frac{13}{6}\) (c) \(\frac{25}{9}\) (d) \(\frac{31}{18}\)

The solution of the differential equation \(\mathrm{ydx}-\left(x+2 y^{2}\right) \mathrm{dy}\) \(=0\) is \(x=f(y)\). If \(f(-1)=1\), then f(a) is equal to (a) 4 (b) 3 (c) 1 (d) 2

If \(y(x)\) is the solution of the differential equation \((x+2) \frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}+4 x-9, x \neq-2\) and \(y(0)=0\), then \(y(-4)\) is equal to: (a) 0 (b) 2 (c) (d) \(-1\)

Let the population of rabbits surviving at time \(t\) be governed by the differential equation \(\frac{d p(t)}{d t}=\frac{1}{2} p(t)-200\). If \(p(0)=100\), then \(p(t)\) equals: (a) \(600-500 e^{1 / 2}\) (b) \(400-300 e^{-t / 2}\) (c) \(400-300 e^{t / 2}\) (d) \(300-200 e^{-i / 2}\)

If the general solution of the differential equation \(y^{\prime}=\frac{y}{x}+\Phi\left(\frac{x}{y}\right)\), for some function \(\Phi\), is given by \(y \ln |c x|=x\), where \(c\) is an arbitrary constant, then \(\Phi(2)\) is equal to: (a) 4 (b) \(\frac{1}{4}\) (c) \(-4\) (d) \(-\frac{1}{4}\)

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers \(x\) is given by \(\frac{d P}{d x}=100-12 \sqrt{x}\). If the firm employs 25 more workers, then the new level of production of items is (a) 2500 (b) 3000 (c) 3500 (d) 4500

If a curve passes through the point \(\left(2, \frac{7}{2}\right)\) and has slope \(\left(1-\frac{1}{x^{2}}\right)\) at any point \((x, y)\) on it, then the ordinate of the point on the curve whose abscissa is \(-2\) is (a) \(-\frac{3}{2}\) (b) \(\frac{3}{2}\) (c) \(\frac{5}{2}\) (d) \(-\frac{5}{2}\)

Consider the differential equation: \(\frac{d y}{d x}=\frac{y^{3}}{2\left(x y^{2}-x^{2}\right)}\) Statement-1: The substitution \(z=y^{2}\) transforms the above equation into a first order homogenous differential equation. Statement-2: The solution of this differential equation is \(y^{2} e^{-y^{2} / x}=C\) (a) Both statements are false. (b) Statement- 1 is true and statement- 2 is false. (c) Statement- 1 is false and statement- 2 is true. (d) Both statements are true.

The population \(p(t)\) at time t of a certain mouse species satisfies the differential equation \(\frac{d p(t)}{d t}=0.5 \mathrm{p}(\mathrm{t})-450\). If \(p(0)=850\), then the time at which the population becomes zero is: (a) \(2 \ln 18\) (b) \(\ln 9\) (c) \(\frac{1}{2} \ln 18\) (d) \(\ln 18\)

Let \(y(x)\) be a solution of \(\frac{(2+\sin x)}{(1+y)} \frac{d y}{d x}=\cos x\). If \(y(0)=2\), then \(y\left(\frac{\pi}{2}\right)\) equals (a) \(\frac{5}{2}\) (b) 2 (c) \(\frac{7}{2}\) (d) 3

The curve that passes through the point \((2,3)\), and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact is given by: (a) \(2 y-3 x=0\) (b) \(y=\frac{6}{x}\) (c) \(x^{2}+y^{2}=13\) (d) \(\left(\frac{x}{2}\right)^{2}+\left(\frac{y}{3}\right)^{2}=2\)

If \(\frac{d y}{d x}=y+3>0\) and \(y(0)=2\), then \(y(\ln 2)\) is equal to: [2011] (a) 5 (b) 13 (c) \(-2\) (d) 7

The normal to a curve at \(P(x, y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\), then the curve is a (a) circle (b) hyperbola (c) ellipse (d) parabola.

If \(x \frac{d y}{d x}=y(\log y-\log x+1)\), then the solution of the equation is \(\quad[\mathbf{2 0 0 5}]\) (a) \(y \log \left(\frac{x}{y}\right)=c x\) (b) \(x \log \left(\frac{y}{x}\right)=c y\) (c) \(\log \left(\frac{y}{x}\right)=c x\) (d) \(\log \left(\frac{x}{y}\right)=c y\)

The solution of the equation \(\frac{d^{2} y}{d x^{2}}=e^{-2 x} \quad[2002]\) (a) \(\frac{e^{-2 x}}{4}\) (b) \(\frac{e^{-2 x}}{4}+c x+d\) (c) \(\frac{1}{4} e^{-2 x}+c x^{2}+d\) (d) \(\frac{1}{4} e^{-4 x}+c x+d\)

Let \(y=y(x)\) be the solution of the differential equation \(\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x, x \in\left(0, \frac{\pi}{2}\right)\) If \(y(\pi / 3)=0\), then \(y(\pi / 4)\) is equal to: (a) \(2-\sqrt{2}\) (b) \(2+\sqrt{2}\) (c) \(\sqrt{2}-2\) (d) \(\frac{1}{\sqrt{2}}-1\)