Summary Differential Equations Cheat Sheet

ODE Cheat Sheet Nonhomogeneous Problems Series Solutions
Method of Undetermined Coefficients Taylor Method
First Order Equations f (x) yp (x) P∞
f (x) ∼ n c x , cn =
f (n) (0)
an xn + · · · + a1 x + a0 A n xn + · · · + A 1 x + A 0 n=0 n n!
Separable ae bx Aebx 1. Differentiate DE repeatedly.
y 0 a cos ωx + b sin ωx A cos ωx + B sin ωx
R (x)dy
= f (x)g(y)
R 2. Apply initial conditions.
g(y)
= f (x) dx + C Modified Method of Undetermined Coefficients: if any
term in the guess yp (x) is a solution of the homogeneous 3. Find Taylor coefficients.
Linear First Order equation, then multiply the guess by xk , where k is the 4. Insert coefficients into series form for y(x).
y 0 (x) + p(x)y(x) smallest positive integer such that no term in xk yp (x) is a
R x = f (x) solution of the homogeneous problem. Power Series Solution
µ(x) = exp p(ξ) dξ Integrating factor. P∞
(µy)0 = f µ Exact Derivative. 1. Let y(x) = c (x
n=0 n
− a)n .
R
Reduction of Order
1
Solution: y(x) = µ(x) f (ξ)µ(ξ) dξ + C 2. Find y 0 (x), y 00 (x).
Homogeneous Case
Exact 3. Insert expansions in DE.
Given y1 (x) satisfies L[y] = 0, find second linearly independent
0 = M (x, y) dx + N (x, y) dy solution as v(x) = v(x)y1 (x). z = v 0 satisfies a separable ODE. 4. Collect like terms using reindexing.
Solution: u(x, y) = const where
Condition: My = Nx Nonhomogeneous Case 5. Find recurrence relation.
du = ∂u
∂x
dx + ∂u∂y
dy
∂u
= M (x, y), ∂u
= N (x, y) 6. Solve for coefficients and insert in y(x) series.
∂x ∂y Given y1 (x) satisfies L[y] = 0, find solution of L[y] = f as
v(x) = v(x)y1 (x). z = v 0 satisfies a first order linear ODE.
Non-Exact Form Ordinary and Singular Points
µ(x, y) (M (x, y) dx + N (x, y) dy) = du(x, y) Method of Variation of Parameters y 00 + a(x)y 0 + b(x)y = 0. x0 is a
My = Nx yp (x) = c1 (x)y1 (x) + c2 (x)y2 (x) Ordinary point: a(x), b(x) real analytic in |x − x0 | < R
c01 (x)y1 (x) + c02 (x)y2 (x) = 0 Regular singular point: (x − x0 )a(x), (x − x0 )2 b(x) have


N ∂µ
∂x
− M ∂µ
∂y
= µ ∂M ∂y
− ∂N∂x
.
f (x)
c01 (x)y10 (x) + c02 (x)y20 (x) = a(x) convergent Taylor series about x = x0 .
Special cases
M −N R Irregular singular point: Not ordinary or regular singular
If yM x = h(y), then µ(y) = exp h(y) dy point.
My −Nx R Applications
If N
= −h(x), then µ(y) = exp h(x) dx
Free Fall Frobenius Method
P∞
Second Order Equations x00 (t) = −g 1. Let y(x) = c (x
n=0 n
− x0 )n+r .
v 0 (t) = −g + f (v)
Linear 2. Obtain indicial equation r(r − 1) + a0 r + b0 .
a(x)y 00 (x) + b(x)y 0 (x) + c(x)y(x) = f (x) Population Dynamics 3. Find recurrence relation based on types of roots of
y(x) = yh (x) + yp (x) P 0 (t) = kP (t) indicial equation.
yh (x) = c1 y1 (x) + c2 y2 (x) P 0 (t) = kP (t) − bP 2 (t) 4. Solve for coefficients and insert in y(x) series.
Constant Coefficients Newton’s Law of Cooling
ay 00 (x) + by 0 (x) + cy(x) = f (x) Laplace Transforms
T 0 (t) = −k(T (t) − Ta )
y(x) = erx ⇒ ar2 + br + c = 0 Transform Pairs
Cases Oscillations c
c
Distinct, real roots: r = r1,2 , yh (x) = c1 er1 x + c2 er2 x mx00 (t) + kx(t) = 0 s
1
One real root: yh (x) = (c1 + c2 x)erx mx00 (t) + bx0 (t) + kx(t) = 0 eat , s>a
s−a
Complex roots: r = α ± iβ, yh (x) = (c1 cos βx + c2 sin βx)eαx mx00 (t) + bx0 (t) + kx(t) = F (t) n!
tn , s>0
Types of Damped Oscillation sn+1
ω
Cauchy-Euler Equations Overdamped, b2 > 4mk sin ωt
s2 + ω 2
ax2 y 00 (x) + bxy 0 (x) + cy(x) = f (x) Critically Damped, b2 = 4mk s
cos ωt
y(x) = xr ⇒ ar(r − 1) + br + c = 0 Underdamped, b2 < 4mk s2 + ω 2
a
Cases sinh at
s2 − a2
Distinct, real roots: r = r1,2 , yh (x) = c1 xr1 + c2 xr2 Numerical Methods cosh at
s
One real root: yh (x) = (c1 + c2 ln |x|)xr s2 − a2
Euler’s Method e−as
Complex roots: r = α ± iβ, y0 = y(x0 ), H(t − a) , s>0
s
yh (x) = (c1 cos(β ln |x|) + c2 sin(β ln |x|))xα yn = yn−1 + ∆xf (xn−1 , yn−1 ), n = 1, . . . , N. δ(t − a) e−as , a ≥ 0, s > 0