Chapter 17

The family passing through \((0,0)\) and satisfying the differential equation \(\frac{y_{2}}{y_{1}}=1\) (where \(\left.y_{n}=\frac{d^{n} y}{d x^{n}}\right)\) is (A) \(y=k\) (B) \(y=k x\) (C) \(y=k\left(e^{x}+1\right)\) (D) \(y=k\left(e^{x}-1\right)\)

The solution of differential equation \(\sec ^{2} y \frac{d y}{d x}+2 x \tan y=x^{3}\) is (A) \(\tan y=\frac{1}{2}\left(x^{2}-1\right)+c e^{-x^{2}}\) (B) \(\tan y=\frac{1}{2}\left(x^{2}-1\right)+c e^{x^{2}}\) (C) \(\tan y=\frac{1}{2}\left(x^{2}+1\right)+c e^{-x^{2}}\) (D) \(\tan y=\frac{1}{2}\left(x^{2}+1\right)+c e^{x^{2}}\)

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

The family of curves represented by \(\frac{d y}{d x}=\frac{x^{2}+x+1}{y^{2}+y+1}\) and the family represented by \(\frac{d y}{d x}+\frac{y^{2}+y+1}{x^{2}+x+1}=0\) (A) Touch each other (B) Are orthogonal (C) Are one and the same (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 differential equation \(y d x-x d y+(\log x) d x=0\) is (A) \(y=\log x+c x\) (B) \(y=1+\log x+c\) (C) \(y+c x=\log \frac{1}{x}\) (D) None of these

Solution of \(\left(x^{2} \sin ^{3} y-y^{2} \cos x\right) d x+\left(x^{3} \cos y \sin ^{2} y-2 y \sin x\right)\) \(d y=0\) is (A) \(\frac{x^{3} \sin ^{3} y}{3}=c\) (B) \(x^{3} \sin ^{3} y=y^{2} \sin x+c\) (C) \(\frac{x^{3} \sin ^{3} y}{3}=y^{2} \sin x+c\) (D) None of these

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the \(x\)-intercept of the normal and passing through \((2,1)\) is (A) \(x^{2}+y^{2}-x=0\) (B) \(4 x^{2}+2 y^{2}-9 y=0\) (C) \(2 x^{2}+4 y^{2}-9 x=0\) (D) \(4 x^{2}+2 y^{2}-9 x=0\)

The solution of the differential equation \(\frac{x+y \frac{d y}{d x}}{y-x \frac{d y}{d x}}=x^{2}+2 y^{2}+\frac{y^{4}}{x^{2}}\) is (A) \(\frac{y}{4}+\frac{1}{x^{2}+y^{2}}=c\) (B) \(\frac{y}{x}-\frac{1}{x^{2}+y^{2}}=c\) (C) \(\frac{x}{y}-\frac{1}{x^{2}+y^{2}}=c\) (D) None of these

The solution of the differential equation \((x-y)(2 d y-d x)=3 d x-5 d y\) is (A) \(2 x-y=\log (x-y+z)+c\) (B) \(2 x+y=\log (x-y+z)+c\) (C) \(2 y-x=\log (x-y+z)+c\) (D) None of these

The solution of the differential equation \((x-y)(2 d y-d x)=3 d x-5 d y\) is (A) \(2 x-y=\log (x-y+z)+c\) (B) \(2 x+y=\log (x-y+z)+c\) (C) \(2 y-x=\log (x-y+z)+c\) (D) None of these

Which of the following does not represent the orthogonal trajectory of the system of curves \(\left(\frac{d y}{d x}\right)^{2}\) \(=\frac{a}{x}\) (A) \(9 a(y+c)^{2}=4 x^{3}\) (B) \(y+c=\frac{-2}{3 \sqrt{a}} x^{3 / 2}\) (C) \(y+c=\frac{2}{3 \sqrt{a}} x^{3 / 2}\) (D) All are orthogonal trajectories

Solution of \(\left(\frac{x+y-1}{x+y-2}\right) \frac{d y}{d x}=\left(\frac{x+y+1}{x+y+2}\right)\), given that \(y=1\) when \(x=1\), is (A) \(\log \left|\frac{(x-y)^{2}-2}{2}\right|=2(x+y)\) (B) \(\log \left|\frac{(x-y)^{2}+2}{2}\right|=2(x-y)\) (C) \(\log \left|\frac{(x+y)^{2}+2}{2}\right|=2(x-y)\) (D) None of these

Solution of \((y+x \sqrt{x y}(x+y)+y) d x+(y \sqrt{x y}(x+y)-x) d y=0\) is (A) \(x^{2}+y^{2}=2 \tan ^{-1} \sqrt{\frac{y}{x}}+c\) (B) \(x^{2}+y^{2}=4 \tan ^{-1} \sqrt{\frac{y}{x}}+c\) (C) \(x^{2}+y^{2}=\tan ^{-1} \sqrt{\frac{y}{x}}+c\) (D) None of these

A particle starts at the origin and moves along the \(x\)-axis in such a way that its velocity at the point \((x, 0)\) is given by the formula \(\frac{d x}{d t}=\cos ^{2} \pi x\). Then the particle never reaches the point on (A) \(x=\frac{1}{4}\) (B) \(x=\frac{3}{4}\) (C) \(x=\frac{1}{2}\) (D) \(x=1\)

Solution of \(\frac{d y}{d x}=\frac{y(x \log y-y)}{x(y \log x-x)}\) is (A) \(x^{y}=c y^{x}\) (B) \(x y=c\) (C) \((x y)^{x}=c\) (D) None of these

Solution of \(\frac{x d y}{x^{2}+y^{2}}=\left(\frac{y}{x^{2}+y^{2}}-1\right) d x\) is (A) \(x-\tan ^{-1} \frac{y}{x}=c\) (B) \(\tan ^{-1} \frac{y}{x}=c\) (C) \(x+\tan ^{-1} \frac{y}{x}=c\) (D) None of these

If \(y=y(x)\) and \(\frac{2+\sin x}{y+1}\left(\frac{d y}{d x}\right)=-\cos x, y(0)=1\), then \(y\left(\frac{\pi}{2}\right)=\) (A) \(\frac{1}{3}\) (B) \(\frac{2}{3}\) (C) \(-\frac{1}{3}\) (D) 1

The equation of the curve passing through the point \((1,1)\) and having slope \(\frac{2 a y}{x(y-a)}\) is (A) \(y^{a} \cdot x^{2 a}=e^{y}\) (B) \(y^{a} \cdot x^{2 a}=e^{y-1}\) (C) \(y^{2 a} \cdot x^{a}=e^{y}\) (D) None of these

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

The differential equation of the curve for which the normal at every point passes through a fixed point \((h, k)\) is (A) \(y-k=\frac{d x}{d y}(h-x)\) (B) \(y-k=\frac{d x}{d y}(x-h)\) (C) \(y-k=\frac{d y}{d x}(h-x)\) (D) \(y-k=\frac{d y}{d x}(x-h)\)

The equation of the curve which passes through the point \((2 a, a)\) and for which the sum of the cartesian sub tangent and the abscissa is equal to the constant \(a\), is (A) \(y(x-a)=a^{2}\) (B) \(y(x+a)=a^{2}\) (C) \(x(y-a)=a^{2}\) (D) \(x(y+a)=a^{2}\)

Let \(I\) be the purchase value of an equipment and \(V(t)\) be the value after it has been used for \(t\) years. The value \(V(t)\) depreciates at a rate given by 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) \(e^{-k T}\) (B) \(T^{2}-\frac{I}{k}\) (C) \(I-\frac{k T^{2}}{2}\) (D) \(I-\frac{k(T-t)^{2}}{2}\)

If \(\frac{d y}{d x}=y+3>0\) and \(y(0)=2\), then \(y(\ln 2)\) is equal to (A) \(-2\) (B) 7 (C) 5 (D) 13

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

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}\) determines a family of circles with (A) variable radius and fixed centre (B) variable radius and variable centre (C) fixed radius and variable centre on \(x\)-axis (D) fixed radius and variable centre on \(y\)-axis

The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying the condition \(y(1)=1\) is (A) \(y=\ln x+x\) (B) \(y=x \ln x+x^{2}\) (C) \(y=x e^{(x-1)}\) (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 \(y=c_{1} e^{c_{x} x}\), where \(c_{1}\) and \(c_{2}\) are arbitrary constants is (A) \(y^{\prime}=y^{2}\) (B) \(y^{\prime \prime}=y^{\prime} y\) (C) \(y^{\prime \prime}=y^{\prime}\) (D) \(y y^{\prime \prime}=\left(y^{\prime}\right)^{2}\)

The general solution of the differential equation \(\frac{d y}{d x}+y g^{\prime}(x)=\mathrm{g}(x) \cdot g^{\prime}(x)\) where \(g(x)\) is a given function of \(x\), is (A) \(g(x)+\log [1+y+g(x)]=C\) (B) \(g(x)+\log [1+y-g(x)]=C\) (C) \(g(x)-\log [1+y-g(x)]=C\) (D) None of these

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

Solution of the equation \(\frac{d y}{d x}=e^{x-y}\left(e^{x}-e^{y}\right)\) is (A) \(e^{y}=e^{x}-1+c e^{-e^{t}}\) (B) \(e^{y}=e^{x}-1+c e^{e^{e}}\) (C) \(e^{x}=e^{y}-1+c e^{-e^{\prime}}\) (D) None of these

Solution of the equation \(x\left(\frac{d y}{d x}\right)^{2}+(y-x) \frac{d y}{d x}-y\) \(=0\) is (A) \((x-y+c)(x y-c)=0\) (B) \((x+y+c)(x y-c)=0\) (C) \((x-y+c)(2 x y-c)=0\) (D) \((y-x+c)(x y-c)=0\)

The differential equation of the family of general circles is (A) \(y^{\prime \prime \prime}\left(1+y^{\prime 2}\right)-3 y^{\prime} y^{\prime \prime 2}=0\) (B) \(y^{\prime \prime \prime}\left(1+y^{\prime 2}\right)+3 y^{\prime} y^{\prime \prime 2}=0\) (C) \(y^{\prime \prime \prime}\left(1+y^{\prime 2}\right)-3 y^{\prime \prime} y^{\prime 2}=0\) (D) None of these

The equation of the family of curves which intersect the hyperbola \(x y=2\) orthogonally is (A) \(y=\frac{x^{3}}{6}+C\) (B) \(y=\frac{x^{2}}{4}+C\) (C) \(y=-\frac{x^{3}}{6}+C\) (D) \(y=-\frac{x^{2}}{4}+C\)

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

Solution of the differential equation \(\left(\frac{x+y-1}{x+y-2}\right) \frac{d y}{d x}=\left(\frac{x+y+1}{x+y+2}\right)\), given that \(y=1\) when \(x=1\), is \(k(y-x)+\log \left|\frac{(x+y)^{k}-k}{k}\right|=0\), where \(k=\) (A) 1 (B) 2 (C) 3 (D) 4

Solution of the equation \(x d y-\left[y+x y^{3}(1+\log x)\right]\) \(d x=0\) is (A) \(\frac{-x^{2}}{y^{2}}=\frac{2 x^{3}}{3}\left(\frac{2}{3}+\log x\right)+C\) (B) \(\frac{x^{2}}{y^{2}}=\frac{2 x^{3}}{3}\left(\frac{2}{3}+\log x\right)+C\) (C) \(\frac{-x^{2}}{y^{2}}=\frac{x^{3}}{3}\left(\frac{2}{3}+\log x\right)+C\) (D) None of these

Solution of equation \(\frac{d y}{d x}=\frac{y \frac{d(\phi(x))}{d x}-y^{2}}{\phi(x)}\) is (A) \(y=\frac{\phi(x)+c}{x}\) (B) \(y=\frac{\phi(x)}{x}+c\) (C) \(y=\frac{\phi(x)}{x+c}\) (D) \(y=\phi(x)+x+c\)

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

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the \(x\)-intercept of the normal and passing through \((2,1)\) is (A) \(x^{2}+y^{2}-x=0\) (B) \(4 x^{2}+2 y^{2}-9 y=0\) (C) \(2 x^{2}+4 y^{2}-9 x=0\) (D) \(4 x^{2}+2 y^{2}-9 x=0\)

Solution of \(\frac{x d x+y d y}{x d y-y d x}=\frac{\sqrt{1-\left(x^{2}+y^{2}\right)}}{\sqrt{x^{2}+y^{2}}}\) is (A) \(\sin ^{-1} \sqrt{x^{2}+y^{2}}=c\) (B) \(\tan ^{-1} \frac{y}{x}=c\) (C) \(\sin ^{-1 \sqrt{x^{2}+y^{2}}}=\tan ^{-1 \frac{y}{x}}+c\) (D) None of these

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

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

The equation of the curve, passing through \((2,5)\) and having the area of triangle formed by the \(x\)-axis, the ordinate of a point on the curve and the tangent at the point 5 square units, is (A) \(x y=10\) (B) \(x^{2}=10 y\) (C) \(y^{2}=10 x\) (D) \(x y^{1 / 2}=10\)

If the curve \(y=f(x)\) passing through the point \((1,2)\) and satisfies the differential equation \(x d y+\left(y+x^{3} y^{2}\right)\) \(d x=0\), then (A) \(x y=\frac{1}{2}\) (B) \(x^{3} y=2\) (C) \(\frac{1}{x y}=2\) (D) None of these

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

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

The solution of the equation \(\frac{y+\sin x \cos ^{2}(x y)}{\cos ^{2}(x y)} d x+\left(\frac{x}{\cos ^{2}(x y)}+\sin y\right) d y=0\) is (A) \(\tan (x y)+\cos x-\cos y=c\) (B) \(\tan (x y)-\cos x-\cos y=c\) (C) \(\tan (x y)+\cos x+\cos y=c\) (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 \(y\left(\frac{\pi}{6}\right)=\frac{3 \sqrt{3}}{8}\), is (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