If $\frac{d^{2} y}{d x^{2}}=\sin x+e^{x}$,$y(0)=3$,and $\frac{d y}{d x}$ at $x=0$ is $4$,then find the equation of the curve.

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} dt = \int_0^x y(t) dt, 0 \leq x \leq 3, y \geq 0$,$y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$ is equal to :

The rate of decay of mass of a certain substance at time $t$ is proportional to the mass at that instant. The time during which the original mass of $m_{0}$ grams will be reduced to $m_{1}$ grams is (where $k$ is the constant of proportionality):

$A$ differential equation for the temperature $T$ of a hot body as a function of time,when it is placed in a bath which is held at a constant temperature of $32^{\circ} F$,is given by (where $k$ is a constant of proportionality):

Verify that the given function $y = \sqrt{a^{2} - x^{2}}$,where $x \in (-a, a)$,is a solution of the differential equation $x + y \frac{dy}{dx} = 0$ (where $y \neq 0$).

The rate at which the population of a city increases varies as the population. In a period of $20$ years,the population increased from $4$ lakhs to $6$ lakhs. In another $20$ years,the population will be (in $lakhs$)