Methods In Differential Equations Flashcards Preview

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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

If the particular integral can be written as part of the complimentary function

Multiply the p.i by x