Midterm · Sagot ni Sam

A

DE Terminologies

Compare and contrast, with examples.

1–2

A · DE Terminologies

Dependent vs. Independent Variables

Compare and contrast dependent and independent variables, with examples.

3–4

A · DE Terminologies

Ordinary vs. Partial DE

Compare and contrast ordinary and partial differential equations, with examples.

5–6

A · DE Terminologies

Order vs. Degree

Compare and contrast the order and degree of a DE, with examples.

7–8

A · DE Terminologies

General vs. Particular Solution

Compare and contrast general and particular solutions, with examples.

9–10

A · DE Terminologies

Arbitrary constant, particular constant, parameter

Compare and contrast an arbitrary constant, a particular constant, and a parameter, with examples.

11–12

A · DE Terminologies

Linear vs. Nonlinear DE

Compare and contrast linear and nonlinear DEs, with examples.

13–14

A · DE Terminologies

Homogeneous vs. Non-homogeneous DE

Compare and contrast homogeneous and non-homogeneous DEs, with examples.

B

Order & Degree

Identify the order and degree (solve, if applicable).

15

B · Order & Degree

Powers 2/3 and 1/2

\left(\frac{d^3y}{dx^3}\right)^{2/3}=\sqrt{1+\frac{d^2y}{dx^2}}

16

B · Order & Degree

Powers −2/3, 2/3, 1/3

\left(\frac{d^2y}{dx^2}\right)^{-2/3}+\left(\frac{dy}{dx}\right)^{2/3}=x\left(\frac{d^2y}{dx^2}\right)^{1/3}

17

B · Order & Degree

ln(y'') + y' = x

\ln\!\left(\frac{d^2y}{dx^2}\right)+\frac{dy}{dx}=x

18

B · Order & Degree

Δ²φ = 0

\Delta^2\phi=0

C

Verify the Solutions

Laplace's equation and Bessel's function.

19

C · Verify the Solutions

Laplace: cos x sinh y + sin x cosh y

Show that

u(x,y)=\cos x\sinh y+\sin x\cosh y

is a solution of Laplace's equation

\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0

.

20

C · Verify the Solutions

Bessel function J₀

Bessel's function of index zero is

J_0(x)=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(n!)^2\,2^{2n}}

. Verify that it solves

xy''+y'+xy=0

.

D

Eliminate the Constants

n constants → differentiate n times.

21

D · Eliminate the Constants

y = Ae²ˣ + Bxe²ˣ

y=Ae^{2x}+Bxe^{2x}

22

D · Eliminate the Constants

x = c₁cos ωt + c₂sin ωt

x=c_1\cos\omega t+c_2\sin\omega t\qquad(\omega\text{ is a parameter})

23

D · Eliminate the Constants

y = e²ˣ(C₁cos 3x + C₂sin 3x)

y=C_1e^{2x}\cos3x+C_2e^{2x}\sin3x

24

D · Eliminate the Constants

y = c₁x² + c₂e²ˣ

y=c_1x^2+c_2e^{2x}

E

Families of Curves

Obtain the DE, with graphs.

25

E · Families of Curves

Lines with intercept sum α

Straight lines with algebraic sum of the intercepts fixed as

\alpha

.

Typo in the paper. The value after "fixed as" is blank on the paper, so it is kept as $\alpha$ .

Graph · drag the sliders

x/1.5 + y/2.5 = 1 (1.5 + 2.5 = 4)

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

α (sum of intercepts)fixed · stays in DE4a (x-intercept)arbitrary · eliminated1.50

26

E · Families of Curves

All circles

All circles.

Graph · drag the sliders

(x − 0)² + (y − 0)² = 4

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

h (center x)arbitrary · eliminated0k (center y)arbitrary · eliminated0r (radius)arbitrary · eliminated2

27

E · Families of Curves

All parabolas

All parabolas.

Graph · drag the sliders

parabola · width 0.50 · rotated 0°

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

vertex xarbitrary · eliminated0vertex yarbitrary · eliminated-1width factorarbitrary · eliminated0.50rotation (°)arbitrary · eliminated0

28

E · Families of Curves

Upright parabolas, focal length k

Parabolas with axis parallel to the

y

-axis and with distance from vertex to focus fixed as

k

.

Graph · drag the sliders

(x − 0)² = 2(y + 1)

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

k (vertex → focus)fixed · stays in DE0.50h (vertex x)arbitrary · eliminated0v (vertex y)arbitrary · eliminated-1

29

E · Families of Curves

Lines through (a, b) in QIII

Straight lines passing through a fixed point

(a,b)

in Quadrant 3.

Graph · drag the sliders

y + 1.5 = 0.58(x + 2)

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

a (fixed point x < 0)fixed · stays in DE-2b (fixed point y < 0)fixed · stays in DE-1.50slope angle (°)arbitrary · eliminated30

30

E · Families of Curves

All sideways parabolas

Parabolas with axis parallel to the

x

-axis.

Graph · drag the sliders

(y − 0)² = 2(x − 0)

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

h (vertex x)arbitrary · eliminated0k (vertex y)arbitrary · eliminated0a (focal length, ±)arbitrary · eliminated0.50

F

Solve the DEs

Separable, homogeneous, exact, linear, Bernoulli.

31

F · Solve the DEsSeparable

sin x sin y dx + cos x cos y dy = 0

\sin x\sin y\,dx+\cos x\cos y\,dy=0

32

F · Solve the DEsSeparable

y ln x ln y dx + dy = 0

y\ln x\ln y\,dx+dy=0

33

F · Solve the DEsHomogeneous

(x − y ln y + y ln x)dx + x(ln y − ln x)dy = 0

(x-y\ln y+y\ln x)\,dx+x(\ln y-\ln x)\,dy=0

Typo in the paper. Printed with " $dx$ " twice. The second is taken as $dy$ ; otherwise it isn't a DE.

34

F · Solve the DEsLinear

y' = x − 2y cot 2x

y'=x-2y\cot2x

35

F · Solve the DEsExact

(cos x cos y − cot x)dx − sin x sin y dy = 0

(\cos x\cos y-\cot x)\,dx-\sin x\sin y\,dy=0

36

F · Solve the DEsBernoulli

2x³y' = y(y² + 3x²)

2x^3y'=y(y^2+3x^2)

37

F · Solve the DEsSeparable

(2a² − r²)dr = r³ sin θ dθ

(2a^2-r^2)\,dr=r^3\sin\theta\,d\theta\qquad\text{when }\theta=0,\;r=a

38

F · Solve the DEsSeparable

xy³dx + eˣdy = 0

xy^3\,dx+e^{x}\,dy=0\qquad\text{when }x\to\infty,\;y=\tfrac12

39

F · Solve the DEsHomogeneous

(y − √(x² + y²))dx − x dy = 0

\left(y-\sqrt{x^2+y^2}\right)dx-x\,dy=0\qquad\text{when }x=0,\;y=1

40

F · Solve the DEsLinear

L di/dt + Ri = E (RL circuit)

L\frac{di}{dt}+Ri=E\qquad\text{when }t=i=0;\;L,R,E\text{ constants}

41

F · Solve the DEsExact

3y(x² − 1)dx + (x³ + 8y − 3x)dy = 0

3y(x^2-1)\,dx+(x^3+8y-3x)\,dy=0\qquad\text{when }x=0,\;y=1

42

F · Solve the DEsBernoulli

(y⁴ − 2xy)dx + 3x²dy = 0

(y^4-2xy)\,dx+3x^2\,dy=0\qquad\text{when }x=2,\;y=1