Sagot ni Sam

Families of curves · live

Graph Lab

Every “obtain the DE of the family” item, drawn live. The faint red curves are other members of the family; the bold one is yours to move.

arbitrary sliders are the constants you eliminate. fixed sliders are given values that stay in the DE. The number of arbitrary sliders = the order of the DE.

LT #22

Lines at fixed distance Q from origin

Straight lines at a fixed distance QQ from the origin.

The DE

(y−xy′)2=Q2[1+(y′)2](y-xy')^2=Q^2\left[1+(y')^2\right]
See the derivation →
xy
x·cos 35° + y·sin 35° = 2

Every line is tangent to the dashed circle of radius Q.

LT #23

Circles of radius r tangent to y-axis

Circles with fixed radius rr and tangent to the yy-axis.

The DE

(x−r)2[1+(y′)2]=r2(y′)2  ⟺  (2rx−x2)[1+(y′)2]=r2(x-r)^2\left[1+(y')^2\right]=r^2(y')^2\;\Longleftrightarrow\;(2rx-x^2)\left[1+(y')^2\right]=r^2
See the derivation →
xy
(x − 1.3)² + (y − 1)² = 1.69

Same radius; each circle just touches the y-axis.

LT #24

Circles of radius 1

Circles of radius unity.

The DE

[1+(y′)2]3=(y′′)2\left[1+(y')^2\right]^3=(y'')^2
See the derivation →
xy
(x − 0.5)² + (y − 0.5)² = 1

The radius is locked at 1. Only the center moves (2 constants, so 2nd order).

LT #25

Sideways parabolas, focal length a

Parabolas with axis parallel to the xx-axis and with distance from vertex to focus fixed as aa.

The DE

2a y′′+(y′)3=0(or 2ad2xdy2=1)2a\,y''+(y')^3=0\qquad\left(\text{or }2a\frac{d^2x}{dy^2}=1\right)
See the derivation →
xy
(y − 0.5)² = 2(x + 1)

Same shape (same a), only the vertex moves. The 2nd dot is the focus.

ME #25

Lines with intercept sum α

Straight lines with algebraic sum of the intercepts fixed as α\alpha.

The DE

y=xy′+α y′y′−1  ⟺  (y−xy′)(y′−1)=αy′y=xy'+\frac{\alpha\,y'}{y'-1}\;\Longleftrightarrow\;(y-xy')(y'-1)=\alpha y'
See the derivation →
xy
x/1.5 + y/2.5 = 1 (1.5 + 2.5 = 4)

x-intercept + y-intercept always adds up to α.

ME #26

All circles

All circles.

The DE

[1+(y′)2]y′′′−3y′(y′′)2=0\left[1+(y')^2\right]y'''-3y'(y'')^2=0
See the derivation →
xy
(x − 0)² + (y − 0)² = 4

Three free constants (h, k, r), so 3rd order.

ME #27

All parabolas

All parabolas.

The DE

3 y′′ y(4)−5 (y′′′)2=0(or d2dx2[(y′′)−2/3]=0)3\,y''\,y^{(4)}-5\,(y''')^2=0\qquad\left(\text{or }\frac{d^2}{dx^2}\big[(y'')^{-2/3}\big]=0\right)
See the derivation →
xy
parabola · width 0.50 · rotated 0°

Position (2), width (1) and direction (1) make 4 constants, so 4th order.

ME #28

Upright parabolas, focal length k

Parabolas with axis parallel to the yy-axis and with distance from vertex to focus fixed as kk.

The DE

2k y′′=12k\,y''=1
See the derivation →
xy
(x − 0)² = 2(y + 1)

Same opening for every curve, so y'' = 1/(2k) is the same constant for all of them.

ME #29

Lines through (a, b) in QIII

Straight lines passing through a fixed point (a,b)(a,b) in Quadrant 3.

The DE

(x−a) y′−(y−b)=0(a<0,  b<0)(x-a)\,y'-(y-b)=0\qquad(a<0,\;b<0)
See the derivation →
xy
y + 1.5 = 0.58(x + 2)

A 'star' of lines: the point is fixed, only the slope m changes.

ME #30

All sideways parabolas

Parabolas with axis parallel to the xx-axis.

The DE

y′ y′′′−3 (y′′)2=0y'\,y'''-3\,(y'')^2=0
See the derivation →
xy
(y − 0)² = 2(x − 0)

Now the width is free too, so there are 3 constants and the DE is 3rd order.