Chapter Questions
The differential equation of the family of curves, $x^{2}=4 b(y+b), b \square$ $R$, is:(a) $x(y \square)^{2}=x+2 y y \square$(b) $x(y \square)^{2}=2 y y \square-x$(c) $x y \square=y \square$(d) $x(y \square)^{2}=x-2 y y \square$
The differential equation representing the family of ellipse having foci either on the $\mathrm{x}$-axis or on the $\mathrm{y}$-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) $2 \mathrm{x}^{2}$(c) $2 \mathrm{x}$(d) $\frac{1}{2} x^{2}$
If $x^{2}+y^{2}=1$, then(a) $y y^{\prime \prime}-2\left(y^{\prime}\right)^{2}+1=0$(b) $y y "+\left(y^{\prime}\right)^{2}+1=0$(c) $y y "+\left(y^{\prime}\right)^{2}-1=0$(d) $y y^{\prime \prime}+2\left(y^{\prime}\right)^{2}+1=0$
Consider the family of all circles whose centers lie on the straight line $y=x .$ If this family of circle is represented by the differential equation $P y^{\prime \prime}+Q y^{\prime}+1=0$, where $P, Q$ are functions of $x, y$ and $y^{\prime}$ $\left(\right.$ here $\left.y^{\prime}=\frac{d y}{d x}, y^{\prime \prime}=\frac{d^{2} y}{d x^{2}}\right)$, then which of the following statements is (are) true?(a) $P=y+x$(b) $P=y-x$(c) $P+Q=1-x+y+y^{\prime}+\left(y^{\prime}\right)^{2}$(d) $P-Q=x+y-y^{\prime}-\left(y^{r}\right)^{2}$
A curve passes through the point $\left(1, \frac{\pi}{6}\right)$. Let the slope of the curve at each point $(x, y)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$.Then the equation of the curve is (a) $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\log \mathrm{x}+\frac{1}{2}$(b) $\operatorname{cosec}\left(\frac{y}{x}\right)=\log x+2$(c) $\sec \left(\frac{2 y}{x}\right)=\log x+2$(d) $\cos \left(\frac{2 y}{x}\right)=\log x+\frac{1}{2}$
The differential equation representing the family of curves $y^{2}=2 c$ $(x+\sqrt{c})$, where $c$ is a positive parameter, is of (a) order 1(b) order 2(c) degree 3(d) degree 4
The order of the differential equation whose general solution is given by $y=\left(c_{1}+c_{2}\right) \cos \left(x+c_{3}\right)-c_{4} e^{x+c}$, where $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$, are arbitrary constants, is(a) 5(b) 4(c) 3(d) 2
A normal is drawn at a point $P(x, y)$ of a curve. It meets thex-axis at $Q$. If $P Q$ is of constant length $k$, then show that the differential equation describing such curves is $y \frac{d y}{d x}=\pm \sqrt{k^{2}-y^{2}}$ Find the equation of such a curve passing through $(0, k)$.
If $(a+b x) e^{j \alpha}=x$, then prove that $x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}$