Methods In Differential Equations

Question 1

Reverse Product Rule

Answer 1

By inspection the integral will be the x component of the side with dy/dx multiplied by the y component of the other

Question 2

Where reverse product rule cannot be used

Answer 2

Multiply by the integrating factor (I.F.) e^∫P dx, where P is the coefficient of the undifferentiated y-term

Question 3

Where you have a coefficient of dy/dx and can’t use reverse product rule without an I.F.

Answer 3

Divide everything by that coefficient

Question 4

Auxiliary equation

Answer 4

An equation in which the solutions to a differential equation depend- a quadratic with the coefficients

Question 5

Proving that a solution satisfies a second-order derivative

Answer 5

Differentiate twice, plug in and show that it is equal to 0

Question 6

Two real distinct roots of the auxiliary equation (α, β) (homogenous)

Answer 6

y = Ae^αx + Be^βx

Question 7

Equal real roots of the auxiliary equation (α) (homogenous)

Answer 7

y = (A + Bx)e^αx

Question 8

Complex roots of the form +- ωi (homogenous)

Answer 8

y = Acosωx + Bsinωx

Question 9

Complex roots of the form p +- qi (homogenous)

Answer 9

y = e^px(Acosqx + Bsinqx)

Question 10

Solving non-homogenous second-order differential equations

Answer 10

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.

Question 11

f(x) is a constant then substitute

Answer 11

y as λ

Question 12

f(x) is a linear function then substitute

Answer 12

y as λx + μ

Question 13

f(x) is a quadratic function then substitute

Answer 13

y as λx^2 + μx + ν

Question 14

f(x) is a function pe^kx then substitute

Answer 14

y as λe^kx

Question 15

f(x) is a function pcos/sin(kx) then substitute

Answer 15

y as λsin(kx) + μcos(kx)

Question 16

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

Answer 16

Multiply the p.i by x