Particular Solution of a Differential Equation

Q. If $x^y+y^x=a^b,$ then find $\frac{dy}{dx}$.

Q.

The number of arbitrary constants in the general solution of a differential equation of fourth order is
(a) zero
(b) 2
(c) 3
(d) 4

Q.

The value of $2\left({\sin }^6\theta +{\cos }^6\theta \right)-3\left({\sin }^4\theta +{\cos }^4\theta \right)+1$ is

1. $2$

2. $0$

3. $4$

4. $6$

Q.

If $y={\tan }^{-1}x+{\cot }^{-1}x+{\sec }^{-1}x+{\csc }^{-1}x$, then $\frac{dy}{dx}$ is equal to

1. $\frac{x^2-1}{x^2+1}$

2. $\mathrm{\pi }$

3. $0$

4. $1$

Q.

What is the value of the integral $\int e^{ax}\cos \left(bx\right)dx$?

Q.

The differential coefficient of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$ w.r.t ${\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ is

1. $1$

2. $-1$

3. $0$

4. None of these

Q.

If$y=\cos (3{\cos }^{-1}x)$, then $\frac{d^{3}y}{d{x}^3}$is equal to

1. $24$

2. $27$

3. $3–12x^2$

4. $-24$

Q.

The solution of the differential equation $(3xy+y^2)dx+(x^2+xy)dy=0$ is

1. $x^2(2xy+y^2)=c^2$

2. $x^2(2xy-y^2)=c^2$

3. $x^2(y^2-2x{y}^2)=c^2$

4. none of these

Q.

The number of arbitrary constants in the particular solution of a differential equation of third order is
(a) 3
(b) 2
(c) 1
(d) zero

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $xdy=(y+x^3\cos x)dx$ with $y\left(\pi \right)=0$, then $y\left(\frac{\pi }{2}\right)$ is equal to

1. $\frac{{\pi }^2}{2}+\frac{\pi }{4}$
2. $\frac{{\pi }^2}{2}-\frac{\pi }{4}$
3. $\frac{{\pi }^2}{4}-\frac{\pi }{2}$
4. $\frac{{\pi }^2}{4}+\frac{\pi }{2}$

Q.

The function $f\left(x\right)=\sec \left(\log \left(x+\sqrt{1+x^2}\right)\right)$ is

1. Odd

2. Even

3. Neither odd nor even

4. Constant

Q.

The differential equation of all circles with a centre at the origin is?

Q.

The function $\frac{x-2}{x+1},(x\ne -1)$is increasing on the interval

1. $(–\infty ,0]$

2. $[0,\infty )$

3. $R$

4. None of these

Q. If $x{\log }_e\left({\log }_{e}x\right)-x^2+y^2=4(y>0)$, then $\frac{dy}{dx}$ at $x=e$ is equal to :

1. $\frac{(2e-1)}{2\sqrt{4+e^2}}$
2. $\frac{e}{\sqrt{4+e^2}}$
3. $\frac{(1+2e)}{2\sqrt{4+e^2}}$
4. $\frac{(1+2e)}{\sqrt{4+e^2}}$

Q.

The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y\left(1\right)=1$ is

1. $y=x\log x+x$

2. $y=\log x+x$

3. $y=x\log x+x^2$

4. $y=x{e}^{x-1}$

Q.

Evaluate $\tan \left[\left(\frac{1}{2}\right){\sin }^{-1}\left(\frac{2a}{1+a^2}\right)+\left(\frac{1}{2}\right){\cos }^{-1}\left(\frac{1-a^2}{1+a^2}\right)\right]$=

1. $\frac{2a}{1+a^2}$

2. $\frac{1-a^2}{1+a^2}$

3. $\frac{2a}{1-a^2}$

4. None of these

Q.

$Ify=co{s}^{-1}x,thenfind\frac{d^{2}y}{d{x}^2}intermsofyalone.$

Q.

Let the slope of the tangent line to a curve at any point $P\left(x,y\right)$ be given by $\frac{\left(x{y}^2+y\right)}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of $y$, for which the point $\left(3,y\right)$ lies on the curve, is:

1. $-\frac{18}{11}$

2. $-\frac{18}{19}$

3. $-\frac{4}{3}$

4. $\frac{18}{35}$

Q.

If $\left|\begin{array}{ccc}6\mathrm{i}& -3\mathrm{i}& 1\\ 4& 3\mathrm{i}& -1\\ 20& 3& \mathrm{i}\end{array}\right|=\mathrm{x}+\mathrm{iy}$, then

1. $x=3,y=1$

2. $x=3,y=0$

3. $x=0,y=1$

4. $x=0,y=0$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{\sqrt{2}y}{2{\cos }^{4}x-\cos 2x}=x{e}^{{\tan }^{-1}\left(\sqrt{2}\cot 2x\right)},$ $0<x<\frac{\pi }{2}$ with $y\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{32}.$ If $y\left(\frac{\pi }{3}\right)=\frac{{\pi }^2}{18}{e}^{-{\tan }^{-1_{\left(\alpha \right)}}},$ then the value of $3{\alpha }^2$ is equal to

Q.

∫e^x² dx = ?

Q.

The period of the function $f\left(x\right)=cose{c}^2\left(3x\right)+cot\left(4x\right)$ is:

1. $\frac{\mathrm{\pi }}{3}$

2. $\frac{\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{6}$

4. $\mathrm{\pi }$

Q. If $y=3^{x\log x},$ then $\frac{dy}{dx}=$

1. $3^{x\log x}(1+\log x)\log 3$
2. $3^{x\log x}(1+\log x)$
3. $3^x(1+\log x)\log 3$
4. $(1+\log x)\log 3$

Q. Find $f^{\prime }\left(x\right)$ if $f\left(x\right)=(\sin x{)}^{\sin x}$ for all $0<x<\pi$.

Q. If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}=(\tan x-y){\sec }^{2}x,$ $x\in (-\frac{\pi }{2},\frac{\pi }{2}),$ such that $y\left(0\right)=0,$ then $y(-\frac{\pi }{4})$ is equal to :

1. $e-2$
2. $\frac{1}{2}-e$
3. $2+\frac{1}{e}$
4. $\frac{1}{e}-2$

Q.

If$(3+i)(z+\overline{z})-(2+i)(z-\overline{z})+14i=0$, then $z\overline{z}$is equal to

1. $5$

2. $8$

3. $10$

4. $40$

Q. The equation of family of curves for which the length of the normal is equal to the radius vector is

1. $y^2\pm x^2=k$
   
2. $y^2=kx$
3. $y\pm x=k$
   
4. $Noneofthese$

Q. Let $y=y\left(x\right)$ be solution of the following differential equation $e^y\frac{dy}{dx}-2e^y\sin x+\sin x{\cos }^{2}x=0,$ $y\left(\frac{\pi }{2}\right)=0$. If $y\left(0\right)={\log }_e(\alpha +\beta e^{–2}),$ then $4(\alpha +\beta )$ is equal to

Q. Using differentials, find the approximate values of the following:
(xi) $\cos {61}^{\circ }$, it being given that $\sin {60}^{\circ }=0.86603\text{and}{1}^{\circ }=0.01745radian$.

Q. If $y=3\cos \left(\log x\right)+4\sin \left(\log x\right),$ show that $x^2{y}_2+x{y}_1+y=0$