Any curve passes through the point $(3, -4)$. If the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2y}{x}$,then the equation of the curve is . . . . . . .

Verify that the function $y=c_{1} e^{a x} \cos b x+c_{2} e^{a x} \sin b x,$ where $c_{1}, c_{2}$ are arbitrary constants,is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}-2 a \frac{d y}{d x}+\left(a^{2}+b^{2}\right) y=0$.

$A$ radioactive sample has a half-life period of $10 \text{ days}$. If a sample originally has a mass of $1000 \text{ mg}$,then the mass remaining after $50 \text{ days}$ is:

Let $y = y(x)$ be the solution of the differential equation $x dy - y dx = \sqrt{x^2 - y^2} dx$,$x \geq 1$,with $y(1) = 0$. If the area bounded by the lines $x = 1$,$x = e^{\pi}$,$y = 0$ and the curve $y = y(x)$ is $\alpha e^{2\pi} + \beta$,then the value of $10(\alpha + \beta)$ is equal to ....... .

$A$ spherical metal ball at $80^{\circ} C$ cools in $5 \text{ minutes}$ to $60^{\circ} C$ in a surrounding temperature of $20^{\circ} C$. The temperature of the ball after $20 \text{ minutes}$ is approximately: (in $^{\circ} C$)

$A$ tangent is drawn at any point $P(x, y)$ on a curve, which passes through $(1, 1)$. The tangent cuts the $X$-axis and $Y$-axis at $A$ and $B$ respectively. If $AP:BP = 3:1$, then: