Homogeneous Function

Q.

If $u={\tan }^{-1}\left(\frac{y}{x}\right)$, then by Euler’s theorem the value of $x\frac{\partial u}{\partial y}+y\frac{\partial u}{\partial y}=$

1. $\sin \left(u\right)$

2. $\tan \left(u\right)$

3. $0$

4. $\cos \left(u\right)$

Q.

A second-degree homogeneous equation in $xandy$ always represents.

1. A pair of straight lines

2. A circle

3. A conic section

4. None of the above

Q.

Determine the values of $mandn$ so that the following linear equations have infinite solutions$(2m-1)x+3y-5=0;3x+(n-1)y-2=0$

Q.

The solution of the differential equation $\frac{dy}{dx}=y\tan \left(x\right)-2\sin \left(x\right)$ is

1. $y\sin \left(x\right)=c+\sin \left(2x\right)$

2. $y\cos \left(x\right)=c+\frac{1}{2}\sin \left(2x\right)$

3. $y\cos \left(x\right)=c-\sin \left(2x\right)$

4. $y\cos \left(x\right)=c+\frac{1}{2}\cos \left(2x\right)$

Q.

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution?

Discuss.

Q.

The augmented matrix is given for a system of equations

If the system is consistent; find the general solution: Otherwise state that there is no solution:

$\left[\overline{)\begin{array}{ccc}1& 2& -3\\ 0& 1& 4\\ 0& 0& 0\end{array}}\begin{array}{c}-9\\ 5\\ 1\end{array}\right]$

1. No solution

2. $$\begin{gathered} x_1=-9-2x_2+3x_3 \\ {x}_2\text{is free} \\ {x}_3\text{is free} \end{gathered}$$

3. $$\begin{gathered} x_1=-19+11x_3 \\ {x}_2=5-4x_3 \\ {x}_3\text{is free} \end{gathered}$$

4. $$\begin{gathered} x_1=-19+11x_3 \\ {x}_2=5-4x_3 \\ {x}_3=1 \end{gathered}$$

Q. The general solution of the differential equation $(e^y+1)\cos xdx+e^y\sin xdy=0$ is
(where $c$ is constant of integration)

1. $\left(e^y\right)=c\sin x$
2. $\left(e^y\right)\sin x=c$
3. $(e^y+1)\sin x=c$
4. $(e^y+1)\cos x=c$

Q. Which of the following functions are homogeneous?

1. $f\left(x,y\right)=\frac{x-y}{x^2+y^2}$
2. $f\left(x,y\right)=x^{\frac{1}{3}}{y}^{-\frac{2}{3}}{\tan }^{-1}\left(\frac{x}{y}\right)$
3. $f\left(x,y\right)=x\left(\ln \sqrt{x^2+y^2}-\mathrm{lny}\right)+{\mathrm{ye}}^{\frac{x}{y}}$
4. $f\left(x,y\right)=x\left(\ln \frac{2x^2+y^2}{x}-\ln (x+y)\right]+y^2\tan \left(\frac{x+2y}{3x-y}\right)$

Q.

For the given differential equation find the particular solution satisfying the given conditions.

$\frac{dy}{dx}-\frac{y}{x}+cosec\left(\frac{y}{x}\right)=0,y=0whenx=1.$

Q.

For the differential equations in given question find the general solution.​​​​

$\frac{dy}{dx}+y=1(y\ne 1).$

Q. The given function $x\tan \left(x^2/y\right)+y$ is a homogeneous function.

1. False
2. True

Q. The solution of $\frac{dy}{dx}=\frac{y}{x}+\tan \left(\frac{y}{x}\right)$.

1. $\sin \left(\frac{y}{x}\right)=cx$
2. $\cos \left(\frac{y}{x}\right)=cx$
3. $\sin \left(\frac{y}{x}\right)=cy$
4. $\sec \left(\frac{y}{x}\right)+\tan \left(\frac{y}{x}\right)=xc$

Q. The differential equation whose general solution is given by, y =
$\left(c_{1}cos\right(x+c_2\left)\right)-\left(c_3{e}^{-x+c_4}\right)+\left(c_{5}sinx\right)$, where $c_1,c_2,c_3,c_4,c_5$ are arbitrary constants, is

1. $\frac{d^{4}y}{d{x}^4}-\frac{d^{2}y}{d{x}^2}+y=0$
2. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
3. $\frac{d^{5}y}{d{x}^5}+y=0$
4. $\frac{d^{3}y}{d{x}^3}-\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-y=0$

Q.

Find the values of $pandq$ for which the following system of linear equations has infinite number of solution .

$2x+3y=7;(p+q)x+(2p-q)y=3(p+q-1)$

Q. The general solution of the differential equation ${\sin }^{-1}\left(\frac{dy}{dx}\right)=x+y$ is (where $c$ is a constant of integration)

1. $\sin (x-y)=(x+c)\cos (x+y)+1$
2. $\sin (x-y)=(x+c)\cos (x-y)+1$
3. $\sin (x+y)=(x+c)\cos (x+y)+1$
4. $\sin (x+y)=(x+c)\cos (x-y)+1$

Q. The general solution of differential equation $(x-y)dy+(x+y)dx=dx+dy$ is
(where $C$ is constant of integration )

1. $xy-\frac{y^2}{2}+\frac{x^2}{2}=x+y+C$
2. $xy+\frac{y^2}{2}-\frac{x^2}{2}=x+y+C$
3. $2xy-\frac{y^2}{3}+\frac{x^2}{3}=x+y+C$
4. $xy-\frac{y^2}{3}+\frac{x^2}{3}=x+y+C$

Q. 2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0

Q. Solution of the equation $xdy=(y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$

1. $f\left(\frac{y}{x}\right)=cx$
2. $f\left(\frac{y}{x}\right)=cxy$
3. $f\left(\frac{y}{x}\right)=0$
4. $f\left(\frac{x}{y}\right)=cy$

Q. The solution of the differential equation $e^{-x}(y+1)dy+({\cos }^{2}x+\sin 2x)ydx=0$ subjected to the condition that $y=1$ when $x=0$ is

1. $y+\log y+e^x{\cos }^{2}x=2$
2. $\log (y+1)+e^x{\cos }^{2}x=1$
3. $y+\log y=e^x{\cos }^{2}x$
4. $(y+1)+e^x{\cos }^{2}x=2$

Q. Assertion :The differential equation of all straight lines which are at a constant distance $p$ from the origin is ${(y-x{y}_1)}^2=p^2(1+y_1^2)$ Reason: The general equation of any straight line which is at a constant distance $p$ from the origin is $x\cos \alpha +y\sin \alpha =p.$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. 2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0

Q. Solve the differential equation $(1+e^{2x})dy+(1+y^2)e^{x}dx=0$ given that $y=1$ when $x=0$.

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

1. $xlog\left(\frac{y}{x}\right)=cy$
2. $log\left(\frac{y}{x}\right)=cx$
3. $log\left(\frac{x}{y}\right)=cx$
4. $ylog\left(\frac{x}{y}\right)=cx$

Q. Determine whether or not the following function is homogeneous: $f(x,y)=\sqrt{x^2+2xy+3y^2}$
If homogeneous enter 1 else enter 0.

Q. Solve: $(2x+3y)dx+xdy=0;$ when $x=1,y=0$.

Q. The degree of the homogeneous function $x^{2}sin\left(\frac{y^2}{x^2}\right)+y^2$is ___

Q. Solution of the differential equation $(3x{y}^2+x\sin xy)dy+(y^3+y\sin xy)dx=0$ is:

1. $x{y}^3+\cos xy=c$
2. $x{y}^2+\cos xy=c$
3. $x{y}^2-\cos xy=c$
4. $x{y}^3-\cos xy=c$

Q. If $y=e^{-x}cos2x$ then which of the following differential equation is satisfied?

1. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$
2. $\frac{d^{2}y}{d{x}^2}+5\frac{dy}{dx}+2y=0$
3. $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}-2y=0$
4. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-5y=0$

Q. $xtan\left(x^2/y\right)+y$
The given function is a homogeneous function.

1. True
2. False

Q. The differential equation by eliminating the arbitary constants from the equation $y=$ $a\cos x+b\sin x+x\sin x,$ where $y_1=\frac{dy}{dx},y_2=\frac{d^{2}y}{d{x}^2}$ is

1. $y_2+y=3\cos x$
2. $y_2+2y=2\cos x$
3. $y_2+y=2\cos x$
4. $y_2+y=4\cos x$