ORDINARY DIFFERENTIAL EQUATION REVIEW SHEET I [28/10/2015] EQUATIONS IMPORTANT = udv
uv − v du
d(lnx) = x1 dx d (sin ) = cos(x) x dx d (cos ) = − sin(x) x dx d x (tan ) = sec2 (x) dx d (csc x) = − csc(x) cot( cot(x)x dx d (sec x) = sec(x) tan( tan(x) dx d (cot x) = − csc2 (x) dx f ( g ) = f gg−2fg
• Infinite Many Sol
• Explicit
Dependent variable can be separated, x + 2 y = 0 • Particular
solution, n-parameter family of solution, particular solution.
Directional Fields Autonomous DE
• Ordinary Differential Equation
One or more function with respect to a single independent variable. variable. One or more functions of two or more independent variables.
SEPARABLE SEPARABLE EQUATIONS EQUATIONS dy = g (x)f (y ) dx
2. Integral Integral
Order of DE − 4y = e
2
∂ 4 u ∂x 4
+
1 f (y )
f (y )
dy =
3. Solution: Solution:
x
∂ 2 u ∂t 2
= 0
Dependent variable y and its first degree dy y and dx , ... ... depend at most on the The coefficient of y independet variable x .
Non-Linear DE
g x dx + C
−
• 3x2 − 2 : Ax2 + Bx + C • sin(4x) : A cos(4x) + B sin(4x)
1. Standard Form: Form: y + P (x)y = f (x)
I x y
y
Homogeneous & Non-Homogeneous DE Linear, Separable & Exact DE Solutions of DE th
A solution of n order DE is a function φ that possesses atleast ‘n’ derivative. F (x, φ(x), φ (x),...,φ( n)(x)) = 0 for all x in I .
differential • Trivial: A solution of an nth-order ordinary differential equation equation is a function function ’phi’ that possesses possesses at least ’n’ derivatives. • Implicit
Dependent variable cannot be separated, sin(x + ey ) = 3y
1
I (x)
• cos(4x) : A cos(4x) + B sin(4x)
P (x)dx
I x f x dx + C
I x f x dx + C
Solution: y = y c (homogeneous) + yp (non − homogeneous) Autonomous DE: When a 1 , a0 &g (x) are constants. Initial Value Problem: Has unique solution. Interval: P &f (x) are continuous Singular Points: Values of x for which a 1 (x) = 0
Linear Equations dn y dn−1 y dy SOLVE:an (x) dx + an−1 (x) dx + ...a1 (x) dx + a0 (x)y = g (x) n n−1 n−1 IC: y (x0 ) = y 0 , y (x0 ) = y 1 ,...,y (x0 ) = y n−1
Initial Value Problem • Unique Sol: an (x), an−1 ,...,a0 (x) and g (x) continuous on interval I and a n (x) = 0 for every x in this interval.
Boundary Value Problem
(3x2 − 1)
• 5x + 7 : Ax + B
Property 2. Integrating Factor: I (x) = e
x
−
• 1: A
LINEAR EQUATIONS
+ y 2 = 0
• Complex & Conjugate: y = e ax (c1 cos βx + c 2 sin βx ), where m 1 = α + ιβ ,m2 = α − ιβ
Example: 10, x2 − 5x, 15x + 8 e x, sin(3x) − 5x cos(2x), xe Not Applicable: ln x, 1x , tan(x), arcsin(x)
( ) ( ) 4. Integration: ( ) = ( ) ( ) 5. Solution: Solution: =
+ sin(y) = 0
• Real & Equal: y = c 1 em1 x + c 2 xem2 x
Method of Undetermined Coefficients
( )
3. Differenti Differential al Equation: Equation: (I (x)y) = I (x)f (x)
dy (1 − y ) dx + 2y = e x
• Real & Distinct Roots: y = c 1 em1 x + c2 em2 x
g(x): constant, polynomial, exponential, sin , cos
dy a1 (x) dx + a0 (x)y = g (x) dy , ... ... are derivative dx
Second Order Equations
Particular Solution
dy = g (x)dx
4. Constant Solution:
Linear DE
d2 y dx2 d4 y dx4
1
1. Separate Separate x & y :
• Homogeneous linear DE always posses trivial solution y = 0.
ay + by + cy = 0 Let y = e mx , Auxiliary Equation: am2 + bm + c = 0
Independent variable doesn’t appear explicitly. explicitly. (Dont change dy over time) Autonomous, dx = 1 − y 2 dy Non-autonomous, dx = 0 .2xy
• Partial Differential Equation
• A constant multiple is also a solution of homogeneous
DE.
INITIAL VALUE PROBLEM SOLUTION CURVES
Types of Differential Equations
Order 2: Order 4:
Superposition Principle: y = c 1 y1 (x) + c 2 y2 (x) + ... + ck yk (x) Notes:
• Singular Solution: ???
DEFINITION
dy 3 + 5( dx )
• No Sol
• Families of Solution: One parameter family of
Leibniz Rule:
d2 y dx2
• Unique Sol
Variation of Parameters • Complementary Solution: yc = c 1 .y1 + c 2 .y2 • Wronskian: W (y1 , y2 ) • f (x) from Standard Form: y + P y + qy = f (x) • Determine u 1 & u 2 , where u 1 =
w1 w
& u 2 =
w2 w
• Particular Solution: yp = u 1 .y1 + u 2 .y2 • Complete Solution: y = y c + y p
Green’s Green’s Theorem Theorem yp (x) =
Steps:
x xo
G(x, t)f (t)dt, G (x, t) =
y1 (t)y2 (x)−y1 (x)y2 (t) W (t)
• Complementary Solution: y1 & y 2 • w(t): • Green Function: G(x, t) =
y1 (t).y2 (x)−y1 (x)y2 (t) w(t)
• Particular Solution: yp (x) =
x x0
G(x, t)f (t)dt
Linear System • X = AX + F
• Unique Solution: A(t), F (t) be continuous on interval I that contains t 0
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