Quick Review · Sagot ni Sam

01

Order & Degree

Order

The highest derivative that appears.

Degree

The power of that highest derivative, once the DE is a polynomial in its derivatives.

Before reading the degree, clean the equation up:

Radicals or fractional powers: raise both sides to a power (the LCD of the denominators). Example: $(y''')^{2/3}=\sqrt{1+y''}\to(y''')^4=(1+y'')^3$ .
Negative powers: multiply through. Example: $(y'')^{-1}+\dots\to$ multiply by $y''$ .
Derivative inside ln, sin, e: isolate it and invert. Example: $\ln y''=u\Rightarrow y''=e^{u}$ , and $\sin y'=x\Rightarrow y'=\sin^{-1}x$ .
Careful: a big power on a lower derivative does not count. Only the highest one matters.

02

Linear or Nonlinear?

A DE is linear in $y$ when it can be written as

a_n(x)\,y^{(n)}+\dots+a_1(x)\,y'+a_0(x)\,y=f(x)

Look only at the parts with $y$ . It is nonlinear if you see:

a power other than 1: $y^2$ , $(y')^3$ , $\sqrt{y'}$
a product: $y\,y'$ , $y\,y''$
$y$ inside a function: $\sin y$ , $e^{y}$ , $\cos(x+y)$
$y$ in a denominator: $\dfrac{1}{x+y}$

Coefficients can be any function of $x$ , such as $e^x$ , $x^2$ or $\sin x$ . That's still linear.

03

Verifying a Solution

Differentiate the proposed solution as many times as the DE needs.
Substitute everything into the DE.
Simplify. If LHS = RHS for all $x$ , it's a solution. If not, say so and show why.

Implicit solutions ( $F(x,y)=C$ ): take the differential of both sides ( $dC=0$ ) and show you get the DE back.

04

Eliminating Constants

Rule: $n$ arbitrary constants → differentiate $n$ times → a DE of order $n$ .

Solution looks like	Roots	DE
$Ae^{r_1x}+Be^{r_2x}$	$r_1,\,r_2$	$y''-(r_1+r_2)y'+r_1r_2\,y=0$
$(A+Bx)e^{rx}$	$r,\,r$	$y''-2ry'+r^2y=0$
$e^{\alpha x}(A\cos\beta x+B\sin\beta x)$	$\alpha\pm\beta i$	$y''-2\alpha y'+(\alpha^2+\beta^2)y=0$

Try it · Roots → DE builder

α (real part)3β (imaginary part)2

Solution $y=e^{3x}\left(A\cos 2x+B\sin 2x\right)$

Roots $r=3\pm 2i$

y''-6y'+13y=0

Set α = 3, β = 2 to get Long Test #20; repeated r = 4 gives LT #18.

Mixed types like $c_1x^2+c_2e^{kx}$ : use the determinant (Wronskian). Put $y, y', y''$ in the first column and the two functions and their derivatives in the other columns, then set the determinant to 0 (see LT #21, ME #24).

05

DE of a Family of Curves

Write the equation of the family.
Sort the constants: arbitrary (changes from curve to curve) vs. fixed (given: r, a, Q, k, α).
Differentiate once per arbitrary constant.
Eliminate the arbitrary constants. The fixed ones stay in the answer.

Want to see it? Every family is live in the Graph Lab.

06

Which Solving Method?

Separable

ME 31, 32, 37, 38

Recognize

Can be arranged as

f(x)\,dx+g(y)\,dy=0

Do

Integrate each side:

\int f\,dx+\int g\,dy=C

Homogeneous

ME 33, 39

Recognize

M

,

N

homogeneous of the same degree (depends only on

y/x

)

Do

Let

y=vx

,

dy=v\,dx+x\,dv

, then separate

Exact

ME 35, 41

Recognize

\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}

Do

F=\int M\,dx+g(y)

; find

g

from

F_y=N

Linear

ME 34, 40

Recognize

y'+P(x)\,y=Q(x)

Do

IF

=e^{\int P\,dx}

;

\;y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C

Bernoulli

ME 36, 42

Recognize

y'+P(x)\,y=Q(x)\,y^n

Do

Let

v=y^{1-n}

; it becomes linear in

v

With a condition (e.g. “when x = 0, y = 1”): plug it into the general solution to get $C$ , which gives the particular solution.

07

Answer Key at a Glance

Long Test

#1	PDE, Order 2, Degree 1
#2	Order 1, Degree 1
#3	Order 3, Degree 1
#4	Order 2, Degree 1
#5	Order 2, Degree 2
#6	Order 2, Degree 8
#7	Linear
#8	Linear
#9	Nonlinear (√y')
#10	Nonlinear (cos(x+y))
#11	Nonlinear (y in denominator)
#12	Nonlinear (y², yy')
#13	No (as printed); yes with “−”
#14	Yes, it is a solution
#15	No (it solves y'' − 2y' + y = 0)
#16	Yes, it is a solution
#17	Yes, it is a solution
#18	y'' − 8y' + 16y = 0
#19	x'' + ω²x = 0
#20	y'' − 6y' + 13y = 0
#21	(x²+x)y'' + (2x²−1)y' − (4x+2)y = 0
#22	(y − xy')² = Q²[1 + (y')²]
#23	(x − r)²[1 + (y')²] = r²(y')²
#24	[1 + (y')²]³ = (y'')²
#25	2a·y'' + (y')³ = 0

Midterm Exam

#1–2	Dependent = unknown function (y); independent = variable it depends on (x)
#3–4	ODE: 1 independent variable; PDE: 2 or more (∂)
#5–6	Order = highest derivative; degree = its power
#7–8	General = with C (family); particular = C found (one curve)
#9–10	Arbitrary = free; particular = value from IC; parameter = fixed, stays
#11–12	Linear: y, y', … to power 1 only; nonlinear: otherwise
#13–14	Homogeneous: RHS = 0; non-homogeneous: RHS = f(x) ≠ 0
#15	Order 3, Degree 4
#16	Order 2, Degree 3
#17	Order 2, Degree 1
#18	PDE, Order 2, Degree 1
#19	Verified: u_xx + u_yy = 0
#20	Verified: xy'' + y' + xy = 0
#21	y'' − 4y' + 4y = 0
#22	x'' + ω²x = 0
#23	y'' − 4y' + 13y = 0
#24	(x²−x)y'' − (2x²−1)y' + (4x−2)y = 0
#25	(y − xy')(y' − 1) = αy'
#26	[1 + (y')²]y''' − 3y'(y'')² = 0
#27	3y''y⁽⁴⁾ − 5(y''')² = 0
#28	2k·y'' = 1
#29	(x − a)y' = y − b
#30	y'y''' − 3(y'')² = 0
#31	sin y = C cos x
#32	x ln x − x + ln(ln y) = C
#33	x ln x + y ln(y/x) − y = Cx
#34	y = ¼ − (x/2)cot 2x + C csc 2x
#35	sin x cos y − ln\|sin x\| = C
#36	y² = x³/(C − x)
#37	a²/r² + ln(r/a) = cos θ
#38	1/y² = 4 − 2(x+1)e^{−x}
#39	x² = 4 − 4y
#40	i = (E/R)(1 − e^{−Rt/L})
#41	x³y − 3xy + 4y² = 4
#42	x² = y³(x + 2)