Solved Examples

1. The solution of $x^3{\displaystyle \frac{dy}{dx}}+4x^2\tan y=e^x$ secy satisfying $y$ ( 1$)=0$, is

(a) $\tan y=(x-2){\mathrm{e}}^x\cdot {\log }_{\mathrm{e}}\mathrm{x}$

(b) $\sin y=e^x(x-1)x^{-4}$

(c) $\tan y=(\mathrm{x}-1){\mathrm{e}}^{\mathrm{x}}{\mathrm{x}}^{-3}$

(d) $\sin y=e^x(x-1)x^{-3}$

2. The solution of the equation

${\displaystyle \frac{dy}{dx}}+x(x+y)=x^3(x+y{)}^3-1$ is

(a) ${\displaystyle \frac{1}{x+y}}=x^2+1+{\mathrm{Ce}}^x$

(b) ${\displaystyle \frac{1}{(x+y{)}^2}}=x^2+1+{\mathrm{Ce}}^{{\mathrm{x}}^2}$

(c) ${\displaystyle \frac{1}{(x+y{)}^2}}=x+1+C{e}^x$

(d) ${\displaystyle \frac{1}{x+y}}=x+1+{\mathrm{Ce}}^{x^2}$

3. The solution of differential equation

$x=1+xy{\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{x^2{y}^2}{2!}}{\left({\displaystyle \frac{dy}{dx}}\right)}^2+{\displaystyle \frac{x^3{y}^3}{3!}}{\left({\displaystyle \frac{dy}{dx}}\right)}^3+$ is

(a) $y={\log }_{e}x+C$

(b) $y={({\log }_{\mathrm{e}}\mathrm{x})}^2+\mathrm{C}$

(c) $\mathrm{y}=\pm {\log }_{\mathrm{e}}\mathrm{x}+\mathrm{C}$

(d) $xy=x^y+C$

4. The function $\mathrm{f}(\mathrm{x})$ satisfying $(\mathrm{f}(\mathrm{x}){)}^2+4\mathrm{f}(\mathrm{x})\cdot {\mathrm{f}}^{\prime }(\mathrm{x})+{({\mathrm{f}}^{\prime }(\mathrm{x}))}^2=0$ is given by

(a) ${\mathrm{ke}}^{(2+\sqrt{3})\mathrm{x}}$

(b) ${\mathrm{ke}}^{(4\pm \sqrt{5})\mathrm{x}}$

(c) ${\mathrm{ke}}^{(-2\pm \sqrt{3})\mathrm{x}}$

(d) None of these

5. A curve $f(x)$ has normal at the point $P(1,1)$ given by $a(y-1)+(x-1)=0$. If the slope of tangent at any point on the curve is proportional to the ordinate at that point, then the curve is

(a) $y=e^{ax}-1$

(b) $\mathrm{y}-1={\mathrm{e}}^{\mathrm{ax}}$

(c) $y=e^{a(x-1)}$

(d) None of these

6. If the length of portion of normal intercepted between the curve \& $x$-axis varies as the square of the ordinate, then the curve is

(a) $\mathrm{kx}+\sqrt{1-{\mathrm{k}}^2{\mathrm{x}}^2}=$ C. ${\mathrm{e}}^{\mathrm{ky}}$

(b) $\mathrm{ky}+\sqrt{{\mathrm{k}}^2{\mathrm{y}}^2-1}=\mathrm{C}\cdot {\mathrm{e}}^{\mathrm{kx}}$

(c) $\mathrm{ky}+\sqrt{{\mathrm{k}}^2{\mathrm{x}}^2-1}={\mathrm{Ce}}^{\mathrm{xy}}$

(n) None of these

Exercise

1. Let $y^1(x)+y(x)g^1(x)=g(x)g^1(x),y(0)=0,x\in R$, where $f^1(x)$ denotes ${\displaystyle \frac{d}{dx}}f(x)$ and $g(x)$ is a given non-constant differentiable function on $R$ with $g(0)=g(2)=0$. Then the value of $y(2)$ is

(a) 2

(b) -1

(c) 0

(d) None of these

2. The solution of the differential equation ${\displaystyle \frac{d^{3}y}{d{x}^3}}-8{\displaystyle \frac{d^{2}y}{d{x}^2}}=0$ satisfying $y(0)={\displaystyle \frac{1}{8}},y_1(0)=0$ and $y_2(0)=1$ is $y_1={\displaystyle \frac{e^{8x}-8x+7}{\lambda }}$, then the numerical value of $\lambda$ is

(a) 8

(b) ${\displaystyle \frac{7}{64}}$

(c) 64

(d) ${\displaystyle \frac{-1}{8}}$

3.* If the solution of the differential equation ${\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{\cos x(3\cos y-7\sin x-3)}{\sin y(3\sin x-7\cos y+7)}}=0$ is $(\sin x+\cos y-1{)}^{\lambda }.(\sin x-\cos y+1{)}^{\mu }=c$, where $c$ is arbitrary constant, then

(a) $\lambda =5$

(b) $\lambda =2$

(c) $\mu =2$

(d) $\mu =5$

4.* The curve for which the area of the triangle formed by the $\mathrm{x}$-axis, the tangent line and radius vector of the point of tangency is equal to ${\mathrm{a}}^2$ is

(a) $\mathrm{x}=\mathrm{Cy}+{\displaystyle \frac{{\mathrm{a}}^2}{\mathrm{y}}}$

(b) $y=x-C{x}^2$

(c) $\mathrm{y}=\mathrm{Cx}+{\displaystyle \frac{{\mathrm{a}}^2}{\mathrm{x}}}$

(d) $x=Cy-{\displaystyle \frac{a^2}{y}}$

5.* The solution of $\left({\displaystyle \frac{xdx+ydy}{xdy-ydx}}\right)=\sqrt{{\displaystyle \frac{a^2-x^2-y^2}{x^2+y^2}}}$ is

(a) $\sqrt{x^2+y^2}=\mathrm{asin}(({\tan }^{-1}y/x)+$ constant $)$

(b) $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=\mathrm{acos}(({\tan }^{-1}\mathrm{y}/\mathrm{x})+$ constant $)$

(c) $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=\mathrm{a}(\tan ({\sin }^{-1}\mathrm{y}/\mathrm{x})+$ constant $)$

(d) $y=x\tan ($ constant $+{\sin }^{-1}{\displaystyle \frac{1}{a}}\sqrt{x^2+y^2})$

6. If $f(x)\int (f(x){)}^{-2}dx$ and $f(1)=0$, then $f(x)$ is equal to

(a) $(2(x-1){)}^{1/4}$

(b) $(5(x-2){)}^{1/5}$

(c) $(3(x-1){)}^{1/3}$

(d) None of these

7. Through any point ( $\mathrm{x},\mathrm{y})$ of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divedes the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of

(a) circle

(b) parabola

(c) ellipse

(d) hyperbola

8. Match the following

9. Read the following passage and answer the questions

If any differential equation be in the form $\mathrm{f}({\mathrm{f}}_1(\mathrm{x},\mathrm{y})\mathrm{d}({\mathrm{f}}_1(\mathrm{x},\mathrm{y})+\varphi ({\mathrm{f}}_2(\mathrm{x},\mathrm{y}))\mathrm{d}({\mathrm{f}}_2(\mathrm{x},\mathrm{y}))+\dots \dots =0$, then we can perform term by term integration. For e.g., $\int \sin xyd(xy)+\int \left({\displaystyle \frac{x}{y}}\right)d\left({\displaystyle \frac{x}{y}}\right)=$ $-\mathrm{cosxy}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{x}}{\mathrm{y}}}\right)}^2+\mathrm{C}$

(i) The solution of the differential equation $(x{y}^4+y)dx-xdy=0$ is

(a) ${\displaystyle \frac{{\mathrm{x}}^2}{4}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{x}}{\mathrm{y}}}\right)}^2=\mathrm{C}$

(b) ${\displaystyle \frac{{\mathrm{x}}^4}{4}}+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\mathrm{x}}{\mathrm{y}}}\right)}^3=\mathrm{C}$

(c) ${\displaystyle \frac{{\mathrm{x}}^4}{4}}-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{x}}{\mathrm{y}}}\right)}^2=\mathrm{C}$

(d) ${\displaystyle \frac{x^2}{4}}-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x}{y}}\right)}^2=C$

(ii) Solution of differential equation $(2x\cos y+y^2\cos x)dx+(2y\sin x-x^2\sin y)dy=0$ is

(a) $x^2\cos y+y^2\sin x=C$

(b) $x\cos y-y\sin x=C$

(c) $x^2{\cos }^{2}y+y^2{\sin }^{2}y=C$

(d) None of these

(iii) Solution of differential equation ${\displaystyle \frac{x^2}{{\displaystyle \frac{1}{x^2}}+y^2}}(xdy+ydx)+y^2(xdy-ydx)=0$ is

(a) ${\tan }^{-1}xy-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{y}{x}}\right)}^2=C$

(b) ${\tan }^{-1}xy+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{y}{x}}\right)}^2=C$

(c) ${\tan }^{-1}xy+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{y}{x}}\right)}^2=\mathrm{C}$

(d) None of these

10. Solution of the differentia equation $(e^{x^2}+e^{y^2})$ y ${\displaystyle \frac{dy}{dx}}+e^{x^2}(x{y}^2-x)=0$ is

(a) ${\mathrm{e}}^{{\mathrm{x}}^2}({\mathrm{y}}^2-1)+{\mathrm{e}}^{{\mathrm{y}}^2}=\mathrm{C}$

(b) ${\mathrm{e}}^{{\mathrm{y}}^2}({\mathrm{x}}^2-1)+{\mathrm{e}}^{{\mathrm{x}}^2}=\mathrm{C}$

(c) ${\mathrm{e}}^{{\mathrm{y}}^2}({\mathrm{y}}^2-1)+{\mathrm{e}}^{{\mathrm{x}}^2}=\mathrm{C}$

(d) ${\mathrm{e}}^{{\mathrm{x}}^2}(\mathrm{y}-1)+{\mathrm{e}}^{{\mathrm{y}}^2}=\mathrm{C}$

11. The solution of the differential equation $(x\mathrm{coty}+\log \cos x)dy+(\log \sin y-y\tan x)dx=0$

(a) $(\sin x{)}^y(\cos y{)}^x=C$

(b) $(\sin y{)}^x(\cos x{)}^y=C$

(c) $(\sin x{)}^x(\cos y{)}^y=C$

(d) None of these

12. The solution of differential equation

${\displaystyle \frac{x+y{\displaystyle \frac{dy}{dx}}}{y-x{\displaystyle \frac{dy}{dx}}}}={\displaystyle \frac{x{\cos }^2(x^2+y^2)}{y^3}}$ is

(a) $\tan ({\mathrm{x}}^2+{\mathrm{y}}^2)={\displaystyle \frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}}+C$

(b) $\cot ({\mathrm{x}}^2+{\mathrm{y}}^2)={\displaystyle \frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}}+\mathrm{C}$

(c) $\tan ({\mathrm{x}}^3+{\mathrm{y}}^3)={\displaystyle \frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}}+\mathrm{C}$

(d) $\cot ({\mathrm{x}}^2+{\mathrm{y}}^2)={\displaystyle \frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}}+\mathrm{C}$