Homogeneous Differential Equation

Q.

What is the meaning of a homogeneous equation?

Q.

Let $\lambda \in R.$ the system of linear equations

$\begin{array}{rcl}2x_1-4x_2+\lambda x_3& =& 1\\ x_1-6x_2+x_3& =& 2\\ \lambda x_1-10x_2+4x_3& =& 3\end{array}$

is inconsistent for:

1. exactly two values of$\lambda$

2. exactly one negative value of$\lambda .$

3. every value of $\lambda .$

4. exactly one positive value of $\lambda .$

Q.

What is the integration of $1$?

Q.

$2{\sin }^2\left(\beta \right)+4\cos \left(\alpha +\beta \right)\sin \left(\alpha \right)\sin \left(\beta \right)+\cos 2\left(\alpha +\beta \right)=$

1. $\sin \left(2\alpha \right)$

2. $\cos \left(2\beta \right)$

3. $\cos \left(2\alpha \right)$

4. $\sin \left(2\beta \right)$

Q.

${\tan }^{-1}\left(\frac{1}{2x+1}\right)+{\tan }^{-1}\left(\frac{1}{4x+1}\right)={\tan }^{-1}\frac{2}{x^2}$ has

1. One Solution

2. Two Solutions

3. Three Solutions

4. No Solution

Q. If the solution of the differential equation $\frac{dy}{dx}=\frac{ax+3}{2y+f}$ represents a circle, then the value of 'a' is

1. 2
2. -2
3. 3
4. -4

Q. The slope of the tangent at (x, y) to a curve passing through $(1,\frac{\pi }{4})$ is given $\frac{y}{x}-co{s}^2\left(\frac{y}{x}\right)$

1. $y=ta{n}^{-1}\left(log\left(\frac{e}{x}\right)\right)$
2. $y=xta{n}^{-1}\left(log\left(\frac{x}{e}\right)\right)$
3. $y=xta{n}^{-1}\left(log\left(\frac{e}{x}\right)\right)$
4. None of these

Q. $\frac{dy}{dx}=\frac{x^2+y^2}{2xy}$ is a homogeneous Differential equation because

1. None of the above.
2. It is not a homogenous differential equation.
3. $\frac{(x^2+y^2)}{2xy}$ is a homogenous function.
4. $\frac{(x^2+y^2)}{2xy}$ is a homogenous function of degree zero.

Q.

If $b_{yx}and{b}_{xy}$ are both positive regression coefficients, then,

1. $\left(\frac{1}{b_{yx}}\right)+\left(\frac{1}{b_{xy}}\right)<\left(\frac{2}{r}\right)$

2. $\left(\frac{1}{b_{yx}}\right)+\left(\frac{1}{b_{xy}}\right)>\left(\frac{2}{r}\right)$

3. $\left(\frac{1}{b_{yx}}\right)+\left(\frac{1}{b_{xy}}\right)<\left(\frac{r}{2}\right)$

4. $noneofthese$

Q.

The form of the differential equation of the central conics ${\text{ax}}^2+b{y}^2=1$ is

1. $\text{x = y (}\frac{\mathrm{dy}}{\mathrm{dx}})$

2. $\mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2+\mathrm{xy}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)-\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=0$

3. $\mathrm{x}+\mathrm{y}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)=0$

4. None of these

Q. Find the particular solution of the differential equation $e^x\tan ydx+(2-e^x){\sec }^{2}ydy=0,$ given that $y=\frac{\pi }{4}$ when $x=0$.

Q. $\text{Solve}:{\tan }^{-1}4x+{\tan }^{-1}6x=\frac{\pi }{4}$

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

1. $y=\ln x+x$
2. $y=x\ln x+x^2$
3. $y=x{e}^{x-1}$
4. $y=x\ln x+x$

Q.

The solution of differential equation $(1+x)ydx+(1-y)xdy=0$ is

1. ${\log }_e\left(xy\right)+x–y=C$

2. ${\log }_e\left(\frac{x}{y}\right)+x+y=C$

3. ${\log }_e\left(\frac{x}{y}\right)-x+y=C$

4. ${\log }_e\left(xy\right)-x+y=C$

Q.

If ${\tan }^{-1}\left(x+2\right)+{\tan }^{-1}\left(x-2\right)-{\tan }^{-1}\left(\frac{1}{2}\right)=0$ then one of the value of $x=$

1. $-1$

2. $5$

3. $\frac{1}{2}$

4. $1$

5. $\frac{-1}{2}$

Q.

Integrate the function: $\frac{4x+1}{\sqrt{2x^2+x-3}}$

Q. If the solution of the differential equation $x\frac{dy}{dx}+y=x{e}^x$ be $xy=e^x\varphi \left(x\right)+c$ then $\varphi \left(x\right)$ is equal to

1. $x+1$
2. $x-1$
3. $1-x$
4. $x$

Q. The locus of midpoints of chords of $y^2=4ax$ which subtend a constant angle $\alpha$ at the vertex is

1. 
2. 
3. 
4. none

Q. The solution of the differential equation $x\frac{dy}{dx}=y(logy-logx+1)$ is

1. $y=x{e}^{cx}$
2. $y+x{e}^{cx}=0$
3. $y+e^x=0$
4. $Noneofthese$

Q.

If ${\tan }^{-1}\left(\alpha +i\beta \right)=x+iy$ then $x=$

1. $\left(\frac{1}{2}\right){\tan }^{-1}\frac{\left(2\alpha \right)}{\left(1-{\alpha }^2-{\beta }^2\right)}$

2. $\left(\frac{1}{2}\right){\tan }^{-1}\frac{\left(2\alpha \right)}{\left(1+{\alpha }^2+{\beta }^2\right)}$

3. ${\tan }^{-1}\frac{\left(2\alpha \right)}{\left(1-{\alpha }^2-{\beta }^2\right)}$

4. None of these

Q.

Tangent to a curve intercepts the y-axis at a point $P$. A line perpendicular to this tangent through $P$ passes through another point (1, 0) the differential equation of the curve is

1. $y\frac{dy}{dx}-x{\left(\frac{dy}{dx}\right)}^2=1$

2. $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$

3. $y\frac{dx}{dy}+x=1$

4. None of these

Q.

For curve $y=\left(\frac{x^2}{2}\right)-2x+5$, the rate of change of $y$ is twice the rate of change of $x$, then the value of $x$ is

Q. The curve satisfying the differential equation, $(x^2-y^2)dx+2xydy=0$ and passing through the point $(1,1)$ is :

1. a circle of radius two
2. an ellipse
3. a hyperbola
4. a circle of radius one

Q.

The following system of linear equations has:

:$$\begin{gathered} 7\mathrm{x}+\; 6\mathrm{y}-\; 2\mathrm{z}=\; 0 \\ 3\mathrm{x}+\; 4\mathrm{y}+2\mathrm{z}=\; 0 \\ \mathrm{x}-\; 2\mathrm{y}-\; 6\mathrm{z}=\; 0 \end{gathered}$$

1. infinitely many solutions, $(x,y,z)$ satisfying $y=2z$

2. infinitely many solutions $(x,y,z)$ satisfying $x=2z$

3. no solution

4. only the trivial solution

Q.

Determine order and degree (when defined) of differential equations.
${\left(\frac{ds}{dt}\right)}^4+3s\frac{d^{2}s}{d{t}^2}=0.$

Q.

A first-order and first-degree differential equation in variables $\mathrm{x}$ and $\mathrm{y}$ is characterized by

1. First order of differentiation of $y$ with respect to $x$and first degree of $\frac{dy}{dx}$

2. Order of differentiation of $y$ with respect to $x$ and power of $y$

3. Powers of $x$ and$y$

4. None of the above

Q. $\int x^2{e}^{x^3}.dx$ equals

1. $\frac{1}{3}{e}^{x^2}+C$
   
2. $\frac{1}{2}{e}^{x^3}+C$
3. $\frac{1}{3}{e}^{x^3}+C$
   
4. $\frac{1}{2}{e}^{x^2}+C$

Q. Among the following equations, the one which is not a homogeneous differential equation is .

1. $\frac{dy}{dx}=\frac{y}{x}$
2. $\frac{dy}{dx}=\frac{x^3+y^3}{2x{y}^2}$
3. $\frac{dy}{dx}=x^2+y^2$
4. $\frac{dy}{dx}=\frac{y^{2}sin\left(x/y\right)}{xy}$

Q. $\frac{dy}{dx}=\frac{x^2+y^2}{2xy}$ is a homogeneous Differential equation because

1. $\frac{(x^2+y^2)}{2xy}$ is not a homogenous function.
2. It is not a homogenous differential equation.
3. $\frac{(x^2+y^2)}{2xy}$ is a homogenous function of degree zero.
4. None of the above.

Q. Which of the following is a homogeneous differential equation? A. B. C. D.