Application of Differential Equations

The family of curves that is orthogonal to $xy=c^2$ is -

The orthogonal trajectory of $y^2=bx$ ($b$ being the parameter) is a conic of eccentricity

The orthogonal trajectory of $y^2=4ax$ (where $a$ being parameter) is

The hemispherical tank of radius $2 \mathrm{m}$ is initially full of water and has an outlet of $12{\mathrm{cm}}^2$ cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $v\left(t\right)=0.6\sqrt{2gh\left(t\right)},$ where $v\left(t\right)$ and $h\left(t\right)$ are, respectively, velocity of the flow through the outlet and the height of water level above the outlet at time $t$ and $g$ is the acceleration due to gravity. Find the time it takes to empty the tank.

Find the time required for a cylindrical tank of radius $r$ and height $H$ to empty through a round hole of area $a$ at the bottom. The flow through a hole is according to the law $U\left(t\right)=u\sqrt{2gh\left(t\right)}$, where $v\left(t\right)$ and $h\left(t\right)$ are respectively the velocity of flow through the hole and the height of the water level above the hole at time $t$ and $g$ is the acceleration due to gravity.

A tank initially contains $50$ gallons of fresh water. Brine contains $2$ pounds per gallon of salt, flows into the tank at the rate of $2$ gallons per minute and the mixture kept uniform by stirring runs out at the same rate. If it will take care for the quantity of salt in the tank to increase from $40$ to $80$ pounds (in seconds) is $206\lambda$, then find $\lambda .$

If the population grows at the rate of $5\%$ per year, then the time taken for the population to become double is (Given $\log 2=0·6912$)

The bacteria increases at the rate proportional to the number of bacteria present. If the original number $N_0$ doubles in $4$ hours, then the number of bacteria in $12$ hours will be

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are $27 \mathrm{gms}$ of certain substance and $3$ hours later it is found that $8 \mathrm{gms}$ are left, then the amount left after one more hour is

The curve such that the intercept on the $x$-axis cut-off between the origin, and the tangent at a point is twice the abscissa and passes through the point $\left(2,3\right)$ is

For next two question please follow the same

Consider a tank which initially holds $V_0$ liter of brine that contains $a$ lb of salt. Another brine solution, containing $b$ lb of salt per liter is poured into the tank at the rate of $e$ $\mathrm{L}/\min$ while, simultaneously, the well-stirred solution leaves the tank at the rate of $f$ $\mathrm{L}/\min$. The problem is to find the amount of salt in the tank at any time $t$.
Let $Q$ denote the amount of salt in the tank at any time. The time rate of change of $Q$, $\frac{dQ}{dt}$, equals the rate at which salt enters the tank at the rate of $be$ $\mathrm{lb}/\min$. To determine the rate at which salt leaves the tank, we first calculate the volume of brine in the tank at any time $t$, which is the initial volume $V_0$ plus the volume of brine added $et$ minus the volume of brine removed $ft$. Thus, the volume of brine at any time is
$V_0+et-ft ....\left(a\right)$
The concentration of salt in the tank at any time is $Q/\left(V_0+et-ft\right)$ from which it follows that salt leaves the tank at the rate of $f\left(\frac{Q}{V_0+et-ft}\right)\mathrm{lb}/\min$. Thus, 

$\frac{dQ}{dt}=be-f\left(\frac{Q}{V_0+et-ft}\right) .....\left(b\right)$

or $\frac{dQ}{dt}+\frac{f}{V_0+et-ft}Q=be$

 A $50 \mathrm{L}$ tank initially contains $10 \mathrm{L}$ of fresh water. At $t=0$, a brine solution containing $1 \mathrm{lb}$ of salt per gallon is poured into the tank at the rate of $4 \mathrm{L}/\min$ while the well-stirred mixture leaves the tank at the rate of $2 \mathrm{L}/\min$. Then the amount of time required for overflow to occur is

A tangent drawn to the curve $\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$ at $\mathrm{P}\left(\mathrm{x}, \mathrm{y}\right)$ cuts the $\mathrm{x} \text{\&} \mathrm{y}$ axis at $\mathrm{A} \text{and} \mathrm{B}$ respectively. If $\mathrm{BP} : \mathrm{AP}=3 : 1$ and $\mathrm{f}\left(1\right)=1$, then the differential equation of curve is

The population of a country increases at the rate proportional to the number of inhabitants. Given that the population of country doubles in $30$ years, then in how many years population of country will triple, (given that $\ln 2=0.6931,\ln 3=1.0986)$

Which of the following is the equation of the orthogonal trajectories of the system of parabolas given by $y=a{x}^2$ ?

The equation of curve passing through the point  $P$$(3,4)$ and satisfying the differential equation $y{\left(\frac{dy}{dx}\right)}^2+(x-y)\frac{dy}{dx}=x=0$ can be

$S_1:x-y+1=0$

$S_2:x+y-7=0$

$S_3:x^2+y^2=25$

$S_4:x^2+y^2-5x=0$

If the subtangent at every point of a curve is bisected at $(0,0)$ and the curve passes through the point$(4,2)$, the equation of curve may be:

The eccentricity of a curve for which tangent at point $P$ intersects the $y-$axis at $M$ such that the point of tangency is at an equal distance from $M$ and the origin is $e$, then $2e^2$ is

If the population of a country doubles in $\text{50}$ years, in how many years will it be three times(triple) under the assumption that the rate of increase is proportional to the number of inhabitants is _____.

Statement 1:
The curves $xy=25$ and $x^2-y^2=16$ cut each other at $90°$.

Statement 2: 
The curve $xy=c$ is the orthogonal trajectory of the curve $x^2-y^2=a^2$.

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Which of the following is revalent?

Statement 1:
The curves $xy=25$ and $x^2-y^2=16$ cut each other at $90°$.

Statement 2: 
The curve $xy=c$ is the orthogonal trajectory of the curve $x^2-y^2=a^2$.

Find the equation of the curve passing through $\left(2, 1\right)$ for which the square of the ordinate is twice the product of the abscissa and the intercept of the normal on $x$-axis.