Solving Homogeneous Differential Equations

Q.

The length of the normal to the curve $y=a\cos h\left(\frac{x}{a}\right)$ at any point varies as

1. ordinate

2. abscissa

3. Square of the abscissa

4. square of the ordinate

Q. The solution of differential equation $(x^2+y^2)dx-2xydy=0$ is

1. $|x^2-y^2|=|x|$
2. $|x^2-y^2|=k|x|$, where k is a positive constant.
3. $|x^2-y^2|=k$, where k is a positive constant.
4. $|x^2-y^2|=k|x-y|$, k is a positive constant.

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

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. Let $P\equiv (-1,0),S_1\equiv x^3-y-3x^2-8x-4=0,S_2\equiv 3x^2-y+7x+4=0.$
Which of the following is/are true:

Q. The solution of $\frac{dy}{dx}=\frac{y}{x}+tan\frac{y}{x}$ is

1. $ysin\left(\frac{y}{x}\right)=cx$
2. $ysin\left(\frac{y}{x}\right)=cy$
3. $sin\left(\frac{x}{y}\right)=cx$
4. $sin\left(\frac{y}{x}\right)=cx$

Q. The solution of $x^2\frac{dy}{dx}-xy=1+\cos \frac{y}{x}$ is

1. $\tan \frac{y}{2x}=C-\frac{1}{2x^2}$
2. $\tan \frac{y}{x}=C+\frac{1}{x}$
3. $\cos \left(\frac{y}{x}\right)=1+\frac{C}{x}$
4. $x^2=(C+x^2)\tan \frac{y}{x}$