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Linear (first order (drivative form a 0 dy/dx + a 1 y = Q(x) (Q(x) ≠ 0…
Linear
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
diffrential form
M(x, y)dx + N(x, y)dy = 0
can be expressed : dy/dx = g(y/x))
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
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 !
constant coffecient
a 0d2 y/dx2 + a 1dy/dx+ a 2 y = 0
m^2=d2 y/dx2and m =dy/dx
substituted to give: Auxillary Equation
a 0 m2+ a 1 m + a 2 = 0
must statisfy : Roots 'm'
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F(x) ≠ 0 non homogenous
constant coffection
Homogenous Part
a 0 ( x )d 2 y/dx 2+ a 1 ( x )dy/dx + a 2 ( x) y = 0
F(x)
particular solution
Variation of Parameters
Undetermined Coefficients
similar to term in complemtary func
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differ from complemtary func
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variable coffecient : t y″ + 4 y′ = t 2
standard form : divide by t
Substitute
integrated factor