mathematics-class-12-unit-09-chapter-04-differential-equations-4-8-3zwadk-mebq

### <Success style="font-size:25px"> Differential Equations L-4 </Success>
### <primary style="font-size:24px"> First order differential equation for the family of circles </primary>
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- (11) Consider two circles with unit radii and centres at $(2,0)$ and $(-2,0)$. Denoting the equations of these circles as $S_1=0$ and $S_2=0$, look at the family of all coaxial circles
- $ S_1+\lambda S_2=0$

- Find the first order differential equation for the family (1.37). Sketch the circles using blue pen. Is there an exceptional one? A limiting circle as it were?

- Sketch circles in red that intersect the blue circles orthogonally.
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<footer style = "font-size: 18px"> $\rightarrow$ Differential equation lecture-4 $\rightarrow$ <span style = "color:blue"> First order differential equation for the family of circles </span> $\rightarrow$ Coaxial circles $\rightarrow$ Exercise </footer>

### <Success style="font-size:25px"> Differential Equations L-4 </Success>
### <primary style="font-size:24px"> Coaxial circles </primary>
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- Coaxial circles - A physical interpretation?
- (12) Is there a physical system that gives rise to the family (1.37) namely
- $\left(x^2+y^2-4 x+3\right)+\lambda\left(x^2+y^2+4 x+3\right)=0 $

- For example can you think of equipotential curves in electrostatics?
- (13) Suppose you do succeed in setting up a physical system with equipotential curves as (1.37) what would be the lines of force for the electric field? Maybe you set it up in three dimensions and look at what happens in the $x$ - yplane? Answer can be found on page 107 (problem 2.47) of David. J. Griffiths, Introduction to electrodynamics, Third Edition, Prentice Hall 1999.

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<footer style = "font-size: 18px">Differential equation lecture-4 $\rightarrow$ First order differential equation for the family of circles $\rightarrow$ <span style = "color:blue"> Coaxial circles </span> $\rightarrow$ Exercise $\rightarrow$ Orthogonal trajectories </footer>

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### <primary style="font-size:24px"> Exercise </primary>
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- More exercises on Diff. Equations for families of curves

- (14) Write down the equations for the one parameter family of circles in the first quadrant touching both the coordinate axes. Find the differential equations for this family.

- (15) Write down the family of tangents to the parabola $y^2=8 x$ and find the differential equation for the family of tangents.

- (16) Find the differential equation for the one parameter family of conics
- $\frac{x^2}{9+c}+\frac{y^2}{4+c}=1 .$

- The algebra must be done a bit cleverly to avoid getting into a mess. After differentiating find $1 /(9+c)$ and $1 /(4+c)$, take reciprocals and subtract.
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<footer style = "font-size: 18px">First order differential equation for the family of circles $\rightarrow$ Coaxial circles $\rightarrow$ <span style = "color:blue"> Exercise </span> $\rightarrow$ Orthogonal trajectories $\rightarrow$ Ubiquity of orthogonal trajectories </footer>

### <Success style="font-size:25px"> Differential Equations L-4 </Success>
### <primary style="font-size:24px"> Orthogonal trajectories </primary>
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- If you turn to page 635 of Halliday-Resnick's book as suggested in the last lecture you would be naturally led to the following geometrical configuration of curves in the plane known as Orthogonal systems of curves in the plane

- Let us consider two one parameter families of curves $C_1$ and $C_2$ Each curve belonging to the family $C_1$ intersects EVERY member of the second family $C_2$ orthogonally and vice versa.
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<footer style = "font-size: 18px">Coaxial circles $\rightarrow$ Exercise $\rightarrow$ <span style = "color:blue"> Orthogonal trajectories </span> $\rightarrow$ Ubiquity of orthogonal trajectories $\rightarrow$ Ubiquity of orthogonal trajectories </footer>

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### <primary style="font-size:24px"> Ubiquity of orthogonal trajectories </primary>
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- Other than electromagnetism, orthogonal trajectories appear in
- Fluid Mechanics.
- Orthogonal trajectories appear in optics. The rays are orthogonal to the wave fronts.
- Orthogonal trajectories appear in Mathematical Cartography, which is the mathematical study of maps. Thus calculus plays an important role in Geography! Have you noticed that on the globe, the latitudes and longitudes meet orthogonally? See chapter 8 of the book J. $\mathrm{McCleary,} \mathrm{Geometry} \mathrm{from} \mathrm{a} \mathrm{differential} \mathrm{viewpoint,} \mathrm{Cambridge}$ University Press 1997.

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<footer style = "font-size: 18px">Exercise $\rightarrow$ Orthogonal trajectories $\rightarrow$ <span style = "color:blue"> Ubiquity of orthogonal trajectories </span> $\rightarrow$ Ubiquity of orthogonal trajectories $\rightarrow$ Example </footer>

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### <primary style="font-size:24px"> Ubiquity of orthogonal trajectories </primary>
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- Other than electromagnetism, orthogonal trajectories appear in
- We shall move on to an area within mathematics where orthogonal trajectories abound ! namely the theory of functions of a complex variable - which incidentally is used by Aerospace engineers !!

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<footer style = "font-size: 18px">Orthogonal trajectories $\rightarrow$ Ubiquity of orthogonal trajectories $\rightarrow$ <span style = "color:blue"> Ubiquity of orthogonal trajectories </span> $\rightarrow$ Example $\rightarrow$ Orthogonal trajectories and functions of a complex variable </footer>

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- Let us next take up a famous function called the Zhukovski function. This function is defined on $C-\{0\}$ by
- $ f(z)=z+\frac{1}{z}$

- Let us see what happens when 2 traces a circle $r \cos r+i r \sin t$ where $r$ is fixed and $t$ can be thought of as time. It is an amusing exercise to understand the image curves. Thinking of the image $f(z(t))$ as a point in the Argand diagram, we get the parametrized curve.
- $\left(\left(r+\frac{1}{r}\right) \cos t,\left(r-\frac{1}{r}\right) \sin t\right)$

- Note that this is a family of ellipses in the plane. We shall call this family the Zhukovskii ellipses.
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<footer style = "font-size: 18px">Ubiquity of orthogonal trajectories $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Orthogonal trajectories and functions of a complex variable </span> $\rightarrow$ Zhukovskii function $\rightarrow$ Example of orthogonal system of curves from complex variables </footer>

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### <primary style="font-size:24px"> Zhukovskii function </primary>
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- The Zhukovskii function

- (17) Determine the foci of the Zhukovskii ellipses. For one value of $r$ you do not get an ellipse. Which one?

- (18) Next assume that the point $z(t)$ in the plane traces out a ray namely $z=t(\cos \theta+i \sin \theta)$ where $\theta$ is fixed and $0<t<\infty$. What are the image curves? Check that they are hyperbolas (Zhukovskii hyperbolas). Find the foci of the Zhukovskii hyperbolas. Show that the Zhukovskii hyperbolas cut the Zhukovskii ellipses orthogonally. Again we have found an example of two families of mutually orthogonal trajectories !

- Zhukovskii employed this function in Aerospace engineering. Another application to an entirely different field appears after a little exercise.
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<footer style = "font-size: 18px">Example $\rightarrow$ Orthogonal trajectories and functions of a complex variable $\rightarrow$ <span style = "color:blue"> Zhukovskii function </span> $\rightarrow$ Example of orthogonal system of curves from complex variables $\rightarrow$ Bohlin's theorem </footer>

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- Another example of orthogonal system of curves from complex variables

- (19) Consider the function $f: C \longrightarrow C$ given by $f(z)=z^2$. Show that the image of horizontal lines $t+i c$ are parabolas. How does the focus of the parabola depend on $c$ ?

- (20) Show that the images of vertical lines $a+$ it are also parabolas. How does the focus of the parabola depend on $a$ ?

- (21) Show that each parabola of exercisen.(19) cuts each parabola of exercise (20) orthogonally.

- Figure: Orthogonal families of parabolas

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<footer style = "font-size: 18px">Orthogonal trajectories and functions of a complex variable $\rightarrow$ Zhukovskii function $\rightarrow$ <span style = "color:blue"> Example of orthogonal system of curves from complex variables </span> $\rightarrow$ Bohlin's theorem $\rightarrow$ Kepler's first law from Bohlin's theorem </footer>

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### <primary style="font-size:24px"> Bohlin's theorem </primary>
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- Zhukovskii function in the service of Astronomy! Bohlin's theorem

- It is remarkable that the Zhukovskii function finds applications in celestial mechanics. It is an old result due to Bohlin (1911) that the map $f: \mathrm{C} \longrightarrow \mathrm{C}$ given by
- $ f(w)=w^2$
- transforms the differential equation of planerary motion into the differential equation
- $\frac{d^2 w}{d s^2}+w=0$
- with a rescaled time $s$. The Zhukovskii map and the Zhukovskii ellipses feature in the proof of Bohlin's theorem.

- It follows easily from Bohlin's theorem that the orbits of the planets are ellipses with the centre of attraction at one of the foci - Kepler's first law of planetary motion!

- How Kepler's first law from Bohlin's theorem

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<footer style = "font-size: 18px">Zhukovskii function $\rightarrow$ Example of orthogonal system of curves from complex variables $\rightarrow$ <span style = "color:blue"> Bohlin's theorem </span> $\rightarrow$ Kepler's first law from Bohlin's theorem $\rightarrow$ Kepler's first law from Bohlin's theorem </footer>

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### <primary style="font-size:24px"> Kepler's first law from Bohlin's theorem </primary>
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- Denoting the real and imaginary parts of $z(s)$ by $x(s)$ and $y(s)$ we get the locus
- $\left(\frac{2 x-\left(a^2-b^2\right)}{a^2+b^2}\right)^2+\frac{y^2}{a^2 b^2}=1 .$

- This is an ellipse with one of the foci at the origin

- Proof of theorems of Newton and Bohlin

- Although this is a theorem on differential equations we shall not prove it but references would be given. The theorem was known to Newton (goes under the name of dual laws of attraction) but the analytic form of the result is due to Bohlin.

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<footer style = "font-size: 18px">Bohlin's theorem $\rightarrow$ Kepler's first law from Bohlin's theorem $\rightarrow$ <span style = "color:blue"> Kepler's first law from Bohlin's theorem </span> $\rightarrow$ Differential equations and orthogonal trajectories $\rightarrow$ Differential equations and orthogonal trajectories </footer>

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### <primary style="font-size:24px"> Differential equations and orthogonal trajectories </primary>
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- Differential Equations and orthogonal trajectories
- suppose we are given a one parameter family of carves $C_2$ is it always possible to find an orthogonal system of trajectories $C_2$ ? To understand this problem let us assume that from the system of curves $C_1$ we obtain a first order differential equation

- $ M(x, y) d x+N(x, y) d y=0 $
- the equation giving phase curves of
- $\frac{d x}{d t}=N(x, y), \quad \frac{d y}{d t}=-M(x, y)$
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<footer style = "font-size: 18px">Kepler's first law from Bohlin's theorem $\rightarrow$ Kepler's first law from Bohlin's theorem $\rightarrow$ <span style = "color:blue"> Differential equations and orthogonal trajectories </span> $\rightarrow$ Differential equations and orthogonal trajectories $\rightarrow$ Theorem on orthogonal trajectories </footer>

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### <primary style="font-size:24px"> Differential equations and orthogonal trajectories </primary>
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- The vector $N(x, y) \mathrm{i}-M(x, y) \mathrm{j}$ is tangent to the curves $C_1$.

- Differential Equation of the Orth. Traj.

- The tangent vectors to the orthogonal trajectories must then be
- $ M(x, y) \mathbf{i}+N(x, y) \mathbf{j}$

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<footer style = "font-size: 18px">Kepler's first law from Bohlin's theorem $\rightarrow$ Differential equations and orthogonal trajectories $\rightarrow$ <span style = "color:blue"> Differential equations and orthogonal trajectories </span> $\rightarrow$ Theorem on orthogonal trajectories $\rightarrow$ Orthogonal trajectories contd </footer>

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- The orthogonal trajectories are thus the phase curves of the system
- $\frac{d x}{d t}=M(x, y), \quad \frac{d y}{d t}=N(x, y)$
- and so the differential equation for the orthogonal trajectories is
- $ N(x, y) d x-M(x, y) d y=0 .$

- Theorem on orthogonal trajectories
- Theorem 1
- If a one parameter family of curves is given by the Diff. Eqn
- $ M(x, y) d x+N(x, y) d y=0$
- then the differential equation for the orthogonal trajectories is
- $ N(x, y) d x-M(x, y) d y=0 .$
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<footer style = "font-size: 18px">Differential equations and orthogonal trajectories $\rightarrow$ Differential equations and orthogonal trajectories $\rightarrow$ <span style = "color:blue"> Theorem on orthogonal trajectories </span> $\rightarrow$ Orthogonal trajectories contd $\rightarrow$ Orthogonal trajectories contd </footer>

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- We shall regard $x$ as a function of $y$.

- The solutions are
- $|x|=\operatorname{aexp}\left(y^2\right)$
- where $a$ is a positive constant.
- The method of separating variables involved division by $x$ which is why we get a special solution namely, the constant function $x=0$ not included in (1.50). This can also be seen geometrically by sketching the curves
- $ y=c \exp \left(-x^2\right)$

- All these curves are bell shaped with a horizontal tangent at the point where the curve cuts the $y$-axis. Thus the $y$-axis (the graph of the constant function $x \equiv 0$ ) must be one of the orthogonal trajectory.
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<footer style = "font-size: 18px">Orthogonal trajectories contd $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Exercise </footer>

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- Exercises:
- (22) Determine the orthogonal trajectories for the family of parabolas in the plane
- $ y=k x^2 .$f
- (23) Consider the family of circles touching the $y$-axis at the origin. Find the differential equation for the orthogonal trajectories to this system. We are not in a position to solve the resulting diff. eqn. We shall be able to do that soon. However you know from geometrical consideration what the orthogonal trajectories are !
- (24) Consider the function $f: C-\{0\} \longrightarrow C$ given by $f(z)=1 / z$. Find the images of the vertical lines under this map. Also find the images of the horizontal lines. Relate this to exercise (23).

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Exercise </span> $\rightarrow$ Exercise $\rightarrow$ Thank you </footer>

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- (25) Find the differential equations for the orthogonal trajectories for the one parameter family of coaxial circles (1.37).
- (26) You were asked to find the differential equations for the family
- $frac{x^2}{9+c}+\frac{y^2}{4+c}=1$

- Find the differential equation for the orthogonal trajectories. Do you see anything remarkable? Can you find an explanation for this?

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<footer style = "font-size: 18px">Example $\rightarrow$ Exercise $\rightarrow$ <span style = "color:blue"> Exercise </span> $\rightarrow$ Thank you $\rightarrow$  </footer>