Maths Solve the Following Question.(4 Marks)

$\left( x +2 y ^2\right) \frac{d y}{d x}= y$, when $x =2, y =1$

$y =\sqrt{a \cos (\log x)+b \sin (\log x)}$

$y=3 \cos (\log x)+4 \sin (\log x) ; x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}+y=0$

$y = e ^{ ax } \sin bx ; \frac{d^2 y}{d x^2}-2 a \frac{d y}{d x}+\left(a^2+b^2\right) y=0$

and water is poured into it. If at any instant the water level rises at the rate of $\left(\frac{\pi}{A}\right) cm / sec$,

where A is the area of the water surface at that instant, show that the vessel will be full in 75 seconds.

$\left(1-x^2\right) \frac{d y}{d x}+2 x y=x\left(1-x^2\right)^{\frac{1}{2}}$

$(x+y) \frac{d y}{d x}=1$

$x \frac{d y}{d x}+2 y=x^2 \cdot \log x$

$\left(x^2+3 x y+y^2\right) d x-x^2 d y=0$

$xy \frac{d y}{d x}= x ^2+2 y ^2, y (1)=0$

$y^2-x^2 \frac{d y}{d x}=x y \frac{d y}{d x}$

$\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{X}{y}\right) d y=0$

$x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0$

$\left(1+2 e^{\frac{x}{y}}\right)+2 e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) \frac{d y}{d x}=0$

$\left(x^2+y^2\right) d x-2 x y \cdot d y=0$