Linear Differential Equations of First Order

Q.

Angle between the line joining the origin to the points of intersection of the curves $2x^2+3y^2+10x=0$ and $3x^2+5y^2+16x=0$ is

1. None of these

2. 

3. 

4. 

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\log }_{e}x,(x>1).$ If $2y\left(2\right)={\log }_{e}4-1,$ then $y\left(e\right)$ is equal to:

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

Q. $\frac{dy}{dx}+x\sin y=\ln x$ is a linear differential equation of first order.

1. True
2. False

Q. The solution of the differential equation $(1+y^2)+(x-e^{ta{n}^{-1}y})\frac{dy}{dx}=0$, is

1. $2x{e}^{ta{n}^{-1}y},=e^{2ta{n}^{-1}y}+k$
2. $x{e}^{ta{n}^{-1}y},=e^{ta{n}^{-1}y}+k$
3. $x{e}^{2ta{n}^{-1}y},=e^{-ta{n}^{-1}y}+k$
4. $(x-2)k{e}^{ta{n}^{-1}y}$

Q. If $y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\frac{2x+1}{x}\right)y=e^{-2x},x>0,$ where $y\left(1\right)=\frac{1}{2}{e}^{-2},$ then:

1. $y\left({\log }_{e}2\right)=\frac{{\log }_{e}2}{4}$
2. $y\left({\log }_{e}2\right)={\log }_{e}4$
3. $y\left(x\right)$ is decreasing in $(0,1)$
4. $y\left(x\right)$ is decreasing in $(\frac{1}{2},1)$