# Help with Differential equation

• Hey there
In summary, a differential equation is a mathematical equation that relates a function with its derivatives and is used to model physical phenomena. They are important because they allow for the description and analysis of complex systems, and come in different types such as ordinary, partial, and stochastic. Solving a differential equation depends on its type and complexity, and they have many real-life applications in fields such as physics, engineering, and economics.
Hey there
A hemispherical bowl of radius a has its axis vertical and is full of water. At time t=0 water starts running out of a small hole in the bottom of the bowl so that the depth of water in the bowl at t is x. The rate at which the volume of water is decreasing is proportional to x. Given that the volume of water in the bowl when the depth is x is $$\pi(ax^2-\frac{1}{3}x^3)$$ show that there is a positive constant k such that

$$\pi(2ax-x^2)\dfrac{dx}{dt}=-kx$$
My method
$$-\dfrac{dV}{dt}\propto x$$
$$\dfrac{dV}{dt}=-kx$$
$$\dfrac{dV}{dt}=\dfrac{dV}{dx} \dfrac{dx}{dt}$$
$$V=\pi(ax^2-\frac{1}{3}x^3)$$
$$\dfrac{dV}{dx}=\pi(2ax-x^2)$$
$$\pi(2ax-x^2)\dfrac{dx}{dt}=-kx$$

(ii) Given that the bowl is empty after a time T, show that

$$k=\dfrac{3 \pi a^2}{2T}$$

My method (I'm not sure how to answer this part).

$$\pi(2ax-x^2)\dfrac{dx}{dt}=-kx$$
$$\displaystyle \int (2\pi a-\pi x )dx=\displaystyle \int-k dt$$
$$2\pi ax-\frac{\pi x^2}{2}=-kt + C$$

x=0 when t=T

I'm not sure how to go about showing $$k=\dfrac{3 \pi a^2}{2T}$$with the information in the question.

Can you help me?

Thanks.

Last edited:
There's a mistake in your integration in part (ii). The integral of 2*pi*a*dx is not 2*pi*a. Fix that first. Then use that at t=0, x=a (the hemisphere was initially full) to find the value of C. Once you've found C put x=0 and solve for t.

Last edited by a moderator:
Dick said:
There's a mistake in your integration in part (ii). The integral of 2*pi*a*dx is not 2*pi*a. Fix that first. Then use that at t=0, x=a (the hemisphere was initially full) to find the value of C. Once you've found C put x=0 and solve for t.

I got $$C=2a^2(\pi-\frac{1}{4})$$

So

$$2 \pi ax-\frac{\pi x^2}{2}=-kt+2a^2(\pi -\frac{1}{4})$$

when x=0 t=T

$$kT=2a^2(\pi-\frac{1}{4})$$

$$k=\dfrac{2a^2(\pi-\frac{1}{4})}{T}$$

$$k=\dfrac{2a^2 \pi-\frac{1}{2}a^2}{T}$$

$$k=\dfrac{4a^2 \pi-a^2}{2T}$$

$$k=\dfrac{a^2(4 \pi -1)}{2T}$$

This doesn't equal $$k=\dfrac{3\pi a^2}{2T}$$

Where have I gone wrong?

Last edited:
If I put x=a into ##2 \pi ax-\frac{\pi x^2}{2}##, which is what you did I hope, I don't get ##2a^2(\pi-\frac{1}{4})##.

Dick said:
If I put x=a into ##2 \pi ax-\frac{\pi x^2}{2}##, which is what you did I hope, I don't get ##2a^2(\pi-\frac{1}{4})##.

I simplified it wrong then.

I'll have another look.

Dick said:
If I put x=a into ##2 \pi ax-\frac{\pi x^2}{2}##, which is what you did I hope, I don't get ##2a^2(\pi-\frac{1}{4})##.

Oh my god. This is incredibly simple. I over complicated it I guess. So I have

$$C=\dfrac{3\pi a^2}{2}$$

$$2\pi ax- \dfrac{\pi x^2}{2}=-kx+\dfrac{3 \pi a^2}{2}$$

x=0 when t=T

$$kT=\dfrac{3 \pi a^2}{2}$$

$$k=\dfrac{3 \pi a^2}{2T}$$

Got it.

Yeah, you had the right idea all along. Some simple errors were tripping you up.

## What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is used to model various physical phenomena in fields such as physics, engineering, and economics.

## Why are differential equations important?

Differential equations are important because they allow us to describe and analyze complex systems and phenomena in a mathematical way. They are used in many fields of science and engineering to make predictions and solve problems.

## What are the different types of differential equations?

The main types of differential equations are ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve single-variable functions, while partial differential equations involve multi-variable functions. Stochastic differential equations involve randomness and probability.

## How do I solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some methods include separation of variables, substitution, and using specific formulas for certain types of equations. Numerical methods can also be used to approximate a solution.

## What are some real-life applications of differential equations?

Differential equations have a wide range of real-life applications, including describing the motion of objects, predicting population growth, modeling chemical reactions, and analyzing electrical circuits. They are also used in fields such as economics, biology, and meteorology.

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