Flashcards in Methods In Differential Equations Deck (16)
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1
Reverse Product Rule
By inspection the integral will be the x component of the side with dy/dx multiplied by the y component of the other
2
Where reverse product rule cannot be used
Multiply by the integrating factor (I.F.) e^∫P dx, where P is the coefficient of the undifferentiated y-term
3
Where you have a coefficient of dy/dx and can't use reverse product rule without an I.F.
Divide everything by that coefficient
4
Auxiliary equation
An equation in which the solutions to a differential equation depend- a quadratic with the coefficients
5
Proving that a solution satisfies a second-order derivative
Differentiate twice, plug in and show that it is equal to 0
6
Two real distinct roots of the auxiliary equation (α, β) (homogenous)
y = Ae^αx + Be^βx
7
Equal real roots of the auxiliary equation (α) (homogenous)
y = (A + Bx)e^αx
8
Complex roots of the form +- ωi (homogenous)
y = Acosωx + Bsinωx
9
Complex roots of the form p +- qi (homogenous)
y = e^px(Acosqx + Bsinqx)
10
Solving non-homogenous second-order differential equations
1. Solve a f''(x) + b f'(x) + cy = 0 for the complimentary function as you would a homogenous
2. Use an appropriate substitution and compare coefficients for the particular integral
3. y = C.F. + P.I.
11
f(x) is a constant then substitute
y as λ
12
f(x) is a linear function then substitute
y as λx + μ
13
f(x) is a quadratic function then substitute
y as λx^2 + μx + ν
14
f(x) is a function pe^kx then substitute
y as λe^kx
15
f(x) is a function pcos/sin(kx) then substitute
y as λsin(kx) + μcos(kx)
16
