Chapter 17

The solution of the equation \(\frac{d y}{d x}=\frac{y}{2 y \log y+y-x}\) (A) \(x=y \log y+\frac{c}{y}\) (B) \(y=x \log x+\frac{c}{x}\) (C) \(x=-y \log y+\frac{c}{y}\) (D) None of these

Solution of the equation \(\cos ^{2} x \frac{d y}{d x}-y \tan 2 x=\cos ^{4} x\), when \(|x|<\frac{\pi}{4}\) and \(y\left(\frac{\pi}{6}\right)=\frac{3 \sqrt{3}}{8}\), (A) \(y=\frac{\sin 2 x}{2\left(\tan ^{2} x-1\right)}\) (B) \(y=\frac{\sin 2 x}{2\left(1-\tan ^{2} x\right)}\) (C) \(y=\frac{\sin 2 x}{2\left(1+\tan ^{2} x\right)}\) (D) None of these

The solution of the equation \(\frac{d y}{d x}+x(x+y)=\) \(x^{3}(x+y)^{3}-1\) is (A) \((x+y)^{-3}=c e^{x^{2}}+x^{2}+1\) (B) \((x+y)^{-2}=c e^{x 2}-x^{2}+1\) (C) \((x+y)^{-2}=c e^{x 2}+x^{2}+1\) (D) None of these

The solution of the equation \(\sin y \frac{d y}{d x}=\cos y(1-x\) \(\cos y\) ) is (A) \(\sec y=(1+x)+c e^{x}\) (B) \(\tan y=(1+x)+c e^{x}\) (C) \(\sec y=(1+x)+c e^{-x}\) (D) None of these

The solution of the differential equation \(x\left(y^{2} e^{x y}+e^{x y}\right) d y=y\left(e^{x / y}-y^{2} e^{x y}\right) d x\) is (A) \(x y=\ln \left(e^{y / x}+c\right)\) (B) \(x y=\ln \left(e^{x / y}+c\right)\) (C) \(\frac{y}{x}=\ln \left(e^{x y}+c\right)\) (D) \(\frac{x}{y}=\ln \left(e^{x y}+c\right)\)

Solution of the equation \(x \int_{0}^{x} y(t) d t=(x+1) \int_{0}^{x} t y(t) d t, x>0\) is (A) \(y=\frac{c}{x^{3}} e^{-\frac{1}{x}}\) (B) \(y=\frac{c}{x^{3}} e^{\frac{1}{x}}\) (C) \(y=\frac{c}{x} e^{-\frac{1}{x^{\prime}}}\) (D) \(y=\frac{c}{x} e^{\frac{1}{x^{1}}}\)

The solution of the differential equation \((1+\tan y)(d x-d y)+2 x d y=0\) is (A) \(x(\sin y+\cos y)=\sin y+c e^{-y}\) (B) \(x(\sin y-\cos y)=\sin y+c e^{-y}\) (C) \(x(\sin y+\cos y)=\cos y+c e^{-y}\) (D) None of these

Solution of the differential equation \(2 y \sin x \frac{d y}{d x}=\) \(2 \sin x \cos x-y^{2} \cos x\) satisfying \(y\left(\frac{\pi}{2}\right)=1\) is given by (A) \(y^{2}=\sin x\) (B) \(y=\sin ^{2} x\) (C) \(y^{2}=\cos x+1\) (D) \(y^{2} \sin x=4 \cos ^{2} x\)

The solution of the equation \(y\left(2 x^{2} y+e^{x}\right) d x-\left(e^{x}+y^{3}\right) d y=0\), if \(y(0)=1\), is (A) \(6 e^{x}-4 x^{3} y-3 y^{3}-3 y=0\) (B) \(6 e^{x}+4 x^{3} y-3 y^{3}-3 y=0\) (C) \(6 e^{x}+4 x^{3} y+3 y^{3}-3 y=0\) (D) None of these

If \(y=c_{1} e^{2 x}+c_{2} e^{x}+c_{3} e^{-x}\) satisfies the differential equation \(\frac{d^{3} y}{d x^{3}}+a \frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0\), then \(\frac{a^{3}+b^{3}+c^{3}}{a b c}\) is equal to (A) \(\frac{1}{4}\) (B) \(-\frac{1}{4}\) (C) \(\frac{1}{2}\) (D) \(-\frac{1}{2}\)

If \(g(x)\) be a function defined on \([-1,1]\). If the area of the equilateral triangle with two of its vertices at \((0,0)\) and \((x, g(x))\) is \(\frac{\sqrt{3}}{4}\), then the function is (A) \(g(x)=\pm \sqrt{\left(1-x^{2}\right)}\) (B) \(g(x)=-\sqrt{\left(1-x^{2}\right)}\) (C) \(g(x)=\sqrt{\left(1-x^{2}\right)}\) (D) \(g(x)=\sqrt{\left(1+x^{2}\right)}\)

For a certain curve \(y=f(x)\) satisfying \(\frac{d^{2} y}{d x^{2}}=6 x-4\), \(f(x)\) has a local minimum value 5 when \(x=1\). (A) Equation of the curve is \(y=x^{3}-2 x^{2}+x+5\) (B) \(f(x)\) has a local maximum at \(x=\frac{1}{3}\) (C) Global maximum value of \(f(x)\) is 7 (D) Global minimum value of \(f(x)\) is 5

The solution of the equation \(\frac{d y}{d x}+x=x e^{(n-1) y}\) is (A) \(\frac{1}{(n-1)} \log \left(\frac{e^{(n-1) y}-1}{e^{(n-1) y}}\right)=\frac{x^{2}}{2}+c\) (B) \(e^{(n-1) y}=c e^{(n-1) y+(n-1)^{\frac{x^{2}}{2}}}+1\) (C) \(\log \left(\frac{e^{(n-1) y}-1}{(n-1) e^{(n-1) y}}\right)=n^{2}+c\) (D) \(e^{(n-1) y}=c e^{(n-1) \frac{x^{x}}{2}+x}+1\)

The solution of the equation \(\left[y\left(1+\frac{1}{x}\right)+\cos y\right] d x+[x+\log x-x \sin y] d y=0\) is (A) \(y(x+\log x)-x \cos y=c\) (B) \(y(x+\log x)+x \sin y=c\) (C) \(y(x+\log x)+x \cos y=c\) (D) None of these

The solution of the differential equation \(\left(x^{2} y-2 x y^{2}\right) d x-\left(x^{3}-3 x^{2} y\right) d y=0\) is (A) \(\frac{x}{y}-2 \log x+3 \log y=c\) (B) \(\frac{x}{y}+2 \log x+3 \log y=c\) (C) \(\frac{x}{y}-2 \log x-3 \log y=c\) (D) None of these

The solution of the equation \((x y \sin x y+\cos x y) y d x+(x y \sin x y-\cos x y) x d y=0\) is (A) \(y \sec x y=c x\) (B) \(x \sec x y=c y\) (C) \(x \operatorname{cosec} x y=c y\) (D) None of these

The integrating factor to make the differential equation \(\left(x y^{2}-e^{\frac{1}{x^{2}}}\right) d x-x^{2} y d y=0\) exact is (A) \(\frac{1}{x}\) (B) \(\frac{1}{x^{2}}\) (C) \(\frac{1}{x^{3}}\) (D) \(\frac{1}{x^{4}}\)

The integrating factor to make the differential equation \(\left(y^{4}+2 y\right) d x+\left(x y^{3}+2 y^{4}-4 x\right) d y=0\) exact is (A) \(\frac{1}{y}\) (B) \(\frac{1}{y^{2}}\) (C) \(\frac{1}{y^{3}}\) (D) \(\frac{1}{y^{4}}\)

The solution of the equation \((x-a)\left(\frac{d y}{d x}\right)^{2}+(x-y) \frac{d y}{d x}-y=0\) is (A) \(y=c x+\frac{a c^{2}}{c+1}\) (B) \(y=c x-\frac{a c^{2}}{c+1}\) (C) \(y=c x-\frac{a^{2} c^{2}}{c+1}\) (D) None of these

The solution of the equation \(p^{2} x(x-2)+p(2 y-2 x y-x+2)+y^{2}+y=0\) is (A) \((y+c x+2 c)(y-c x+1)=0\) (B) \((y-c x+2 c)(y+c x+1)=0\) (C) \((y-c x+2 c)(y-c x+1)=0\) (D) \((y-c x+2 c)(y-c x-1)=0\)

The orthogonal trajectory of the family of parabolas \(y^{2}=4 a x\) is (A) \(2 x^{2}+y^{2}=c\) (B) \(x^{2}+2 y^{2}=c\) (C) \(2 x^{2}-y^{2}=c\) (D) None of these

Assertion: The order of the differential equation, of which \(x y=c e^{x}+b e^{-x}+x^{2}\) is a solution, is 2 . Reason: The differential equation is \(x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2=0\)

Assertion: A normal is drawn at a point \(P(x, y)\) of a curve. It meets the \(x\)-axis and the \(y\)-axis in points \(A\) and \(B\), respectively, such that \(\frac{1}{O A}+\frac{1}{O B}=1\), where \(O\) is the origin. The equation of such a curve passing through \((5,4)\) is \((x-1)^{2}+(y-1)^{2}=25\). Reason: \(O A=x+y \frac{d y}{d x}\) and \(O B=\frac{\left(x+y \frac{d y}{d x}\right)}{\frac{d y}{d x}}\)

Assertion: The differential equation of all straight lines which are at a constant distance \(p\) from the origin is \(\left(y-x y_{1}\right)^{2}=p^{2}\left(1+y_{1}^{2}\right)\) Reason: The general equation of any straight line which is at a constant distance \(p\) from the origin is \(x \cos \alpha+y \sin \alpha=p .\)

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 \(\quad\) (A) \(\left(1, \frac{2}{3}\right)\) (B) \((3,1)\) (C) \((3,3)\) (D) \((1,2)\)

The solution of the equation \(\frac{d^{2} y}{d x^{2}}=e^{-2 x}\) is (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) \(\frac{1}{4} e^{-2 x}+c+d\)

The differential equation of all non-vertical lines in a plane is (A) \(\frac{d^{2} y}{d x^{2}}=0\) (B) \(\frac{d^{2} x}{d y^{2}}=0\) (C) \(\frac{d y}{d x}=0\) (D) \(\frac{d x}{d y}=0\)

The degree and order of the differential equation of the family of all parabolas whose axis is \(x\)-axis, are respectively \(\quad\) [2003] (A) 2,1 \(\begin{array}{ll}\text { (B) } 1,2 & \text { (C) } 3,2\end{array}\) (D) 2,3

The solution of the differential equation \(\left(1+y^{2}\right)\left(x-e^{2 \tan ^{-1} y}\right) \frac{d y}{d x}=0\), is \(\quad\) (A) \((x-2)=k e^{-\tan ^{-1} y}\) (B) \(2 x e^{\tan ^{-1}} y=e^{2 \tan ^{-1}}+k\) (C) \(x e^{\operatorname{lan}^{-1} y}=\tan ^{-1} y+k\) (D) \(x e^{2 \tan ^{-1} y}=e^{\tan ^{-1} y}+k\)

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

The solution of the differential equation \(y d x+(x+\) \(\left.x^{2 y}\right) d y=0\) is (A) \(-\frac{1}{x y}=C\) (B) \(-\frac{1}{x y}+\log y=C\) (C) \(\frac{1}{x y}+\log y=C\) (D) \(\log y=C x\)

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: \(\quad\) (A) order 1 , degree 2 (B) order 1 , degree 1 (C) order 1 , degree 3 (D) order 2, degree 2

If \(x \frac{d y}{d x}=y(\log y-\log x+1)\), then the solution of the equation is (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 differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is (A) \(x^{2}=y^{2}+x y \frac{d y}{d x}\) (B) \(x^{2}=y^{2}+3 x y \frac{d y}{d x}\) (C) \(y^{2}=x^{2}+2 x y \frac{d y}{d x}\) (D) \(y^{2}=x^{2}-2 x y \frac{d y}{d x}\)

The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying the condition \(y(1)=1\) is \(\quad\) (C) \(y=x e^{(x-1)}\) (A) \(y=\ln x+x\) (B) \(y=x \ln x+x^{2}\) (D) \(y=x \ln x+x\)

The differential equation of the family of circles with fixed radius 5 units and centre on the line \(y=2\) is (A) \((x-2) y^{\prime 2}=25-(y-2)^{2}\) (B) \((y-2) y^{\prime 2}=25-(y-2)^{2}\) (C) \((y-2)^{2} y^{\prime 2}=25-(y-2)^{2}\) (D) \((x-2)^{2} y^{\prime 2}=25-(y-2)^{2}\)

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) \(y^{\prime}=y^{2}\) (B) \(y^{\prime \prime}=y^{\prime} \mathrm{y}\) (C) \(y y^{\prime}=y^{\prime}\) (D) \(y y^{\prime}=\left(y^{\prime}\right)^{2}\)

Let \(l\) be the purchase value of an equipment and \(V(t)\) be the value of equipment after it has been used for \(t\) years. The value \(V(t)\) depreciates at a rate given by the differential equation \(\frac{d V(t)}{d t}=k(T-t)\), where \(k>0\) is a constant and \(T\) is the total life in years of the equipment. Then, the scrap value \(V(T)\) of the equipment is (A) \(l-\frac{k T^{2}}{2}\) (B) \(l-\frac{k(T-t)^{2}}{2}\) (C) \(e^{-k T}\) (D) \(T^{2}-\frac{l}{k}\)

The population \(p(t)\) at time \(t\) of a certain mouse species satisfies the differential equation \(\frac{d p(t)}{d t}=0.5 p(t)\) \(-450\) with initial condition \(p(0)=850\), then the value of \(t\) for which \(p(t)=0\) is (A) \(2 \ln 18\) (B) \(\ln 9\) (C) \(\frac{1}{2} \ln 18\) (D) \(\ln 18\)

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production \(P\) with respect to 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) 3000 (B) 3500 (C) 4500 (D) 2500

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

Let \(y(x)\) be the solution of the differential equation \((x \log x) \frac{d y}{d x}+y=2 x \log x,(x \geq 1)\). Then \(y(e)\) is equal to: (A) 0 (B) 2 (C) \(2 e\) (D) \(e\)

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 \(f\left(-\frac{1}{2}\right)\) is equal to: (A) \(\frac{4}{5}\) (B) \(-\frac{2}{5}\) (C) \(-\frac{4}{5}\) (D) \(\frac{2}{5}\)