Methods In Differential Equations Flashcards
What is the perfect derivative?
By inspection the integral will be the x component of the side with dy/dx multiplied by the y component of the other
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
Where you have a coefficient of dy/dx and can’t use reverse product rule without an I.F.
Divide everything by that coefficient
Auxiliary equation
An equation in which the solutions to a differential equation depend- a quadratic with the coefficients
Proving that a solution satisfies a second-order homogenous differential equation
Differentiate twice, plug in and show that it is equal to 0
Two real distinct roots of the auxiliary equation (α, β) (homogenous)
y = Ae^αx + Be^βx
Repeated real roots of the auxiliary equation (α) (homogenous)
y = (A + Bx)e^αx
Complex roots of the form +- ωi (homogenous)
y = Acosωx + Bsinωx
Complex roots of the form p +- qi (homogenous)
y = e^px(Acosqx + Bsinqx)
Solving non-homogenous second-order differential equations
- Solve a f’‘(x) + b f’(x) + cy = 0 for the complimentary function as you would a homogenous
- Use an appropriate substitution and compare coefficients for the particular integral
- y = C.F. + P.I.
f(x) is a constant then substitute
PI is C
f(x) is a linear function then substitute
PI is Cx+ D
f(x) is a quadratic function then substitute
PI is Cx^2 + Dx + E
f(x) is a function pe^kx then substitute
y as Ce^kx
f(x) is a function pcos/sin(kx) then substitute
PI is Ccos(kx) + Dsin(kx)
