Differential Equations - Extra practice

Finding the orthogonal family
Consider the family of hyperbolas $xy=c$. Find its orthogonal family of curves.

Solve by reduction of order

$$\frac{y^{\prime \prime }}{y^{\prime }}=x^2$$

Solve particular by reduction of order

$$y^{\prime \prime }\cdot y^{\prime }=x(1+x)$$

with $y(0)=1$ and $y^{\prime }(0)=0$.

Solve by reduction of order

$$\begin{array}{rl}2y{y}^{\prime \prime }& =1+(y^{\prime }{)}^2\\ y{y}^{\prime \prime }& =y^2{y}^{\prime }+(y^{\prime }{)}^2\end{array}$$

Solve the Bernoulli equations

$$\begin{array}{rl}x{y}^{\prime }+y& =x^4{y}^3\\ y^{\prime }+xy& =x{y}^4\end{array}$$

Classify

Nagle 61 classify separable, linear, exact.

Consider the IVP: $t{y}^{\prime }(t)=3y,y(0)=0$. Analyze the functional solutions to this IVP. Extend youanalysis to $t<0$ as well. This Desmos and picture might help you:
https://www.desmos.com/calculator/j0ykjy2r6e

Problem

Water clock
A clock is formed by puncturing the bottom of a bowl with a small aperture, filling with water, and measuring the height of water in the bowl as it slowly empties over many hours. (For example, a ruler can be dipped into the bowl.) Torricelli’s Law says that the velocity of water exiting the aperture is $v_a=\sqrt{2Gh}$ where $G$ is the gravitational constant, and $h$ the height of water in the bowl. (This can be proved using Bernoulli’s Principle and conservation of energy.)

Your goal is to find the shape of the bowl such that the water line decreases at a constant rate, implying that an ordinary (linear) ruler can be used to measure the time according to the water level.

Let $r$ be the radius of the aperture, and $R(z)$ the radius of the bowl at the height $z$. The cross-section of the bowl at height $z$ is a disk (the bowl is given by revolution), and its area is $A(z)=\pi R(z{)}^2$.

• Find a differential equation relating the rate of change of height $z$ of water in the bowl, time $t$, and $R(z)$.
• Solve for $R(z)$ by setting $z^{\prime }(t)$ constant.

Problem

Housing mortgage: 30 years, given downpayment, given value, given interest, given inflation… what is monthly payment.