Linear
Higher
a 0 (x) dn y/dx n +a 1d n−1 y/dx n−1 +...+a nd n y/dx n +y=F(x)
F(x) = 0
first order
drivative form
a 0 dy/dx + a 1 y = Q(x)
Q(x) ≠ 0
1st Order Linear
Non-Homogenuous
D.E.
Exactness
non exact
Integrating
Factor
exact
seek Function
F(x, y)
General Solution
Q(x)=0
1st Order Linear
Homogenuous D.E.
use y = vx
transform sebrable equation
click to edit
diffrential form
M(x, y)dx + N(x, y)dy = 0
can be expressed : dy/dx = g(y/x))
click to edit
F(x) ≠ 0
second
FORM *a 0 d 2 y /dx 2 +a 1dy/ dx +a 2 y=F(x)
Homogenous F(x) = 0 , may be
a 0 ( x )d 2 y/dx 2+ a 1 ( x )dy/dx + a 2 ( x) y = 0
has : Functions as
Coefficients
uses : Reduction of
Order
where Transformation
y = v(x)
reduces : 2nd Order Homogenous var 'v'and 'x'
by letting : w =dv/dx
Normalized Form :d 2 y/dx 2+ P 1 (x)dy/dx+ P 2 (x)y= 0
P 1 (x) =a 1 (x)/a 0 (x) and P 2 (x) =a 2 (x)/ a 0 (x)
check for : Analyticity at
point x 0
Two Linearly Independent Series Solutions
y = C n ( x − x 0 )n = 0∞∑whereC n =f( n) ( x 0 )n !
click to edit
F(x) ≠ 0 non homogenous
constant coffection
click to edit
Homogenous Part
a 0 ( x )d 2 y/dx 2+ a 1 ( x )dy/dx + a 2 ( x) y = 0
click to edit
F(x)
particular solution
Variation of Parameters
Undetermined Coefficients
similar to term in complemtary func
times by coff and x
differ from complemtary func
exponen
times coffe
poynomial
times coff , same degree
sine or cos
both cosine
and sine (of the same frequency) must appear
product of several func
product consist of the corresponding choices of the individual components (that there should be only as many
undetermined coefficients in Y as there are distinct terms)
ex: y″ − 2y′ − 3y = t 3 e 5t cos(3t)
g(t) = t ^3 e 5t cos(3t)
Correct form: Y = (At 3 + Bt 2 + Ct + D) e 5t cos(3t) +
(Et 3 + Ft 2 + Gt + H) e 5t sin(3t)
Wrong form: Y = (At 3 + Bt 2 + Ct + D) E e 5t (F cos(3t) + G sin(3t))
variable coffecient : t y″ + 4 y′ = t 2
standard form : divide by t
Substitute
integrated factor
constant coffecient
a 0d2 y/dx2 + a 1dy/dx+ a 2 y = 0
m^2=d2 y/dx2and m =dy/dx
click to edit
substituted to give: Auxillary Equation
a 0 m2+ a 1 m + a 2 = 0
must statisfy : Roots 'm'
in order to form :Linear Independent
Solutions of form:
y = e^mx
combined as : Linear
Combination
called Complementary
Function
y c