Differential Equations Flashcards
An equation that contains one of more terms involving derivatives.
Differential Equations
An equation containing only one independent variable
Ordinary Differential Equation
An equation containing two or more independent variables
Partial Differential Equation
order of the highest ordered derivative
Order
Highest power of the highest ordered derivative
Degree
The solution has at least one arbitrary constant
General Solution
The solution has no arbitrary constant
Particular Solution
Variable Separable
It is variable if you can rearrange the equation to form
P(x)dx = Q(y)dy
then derive respectively to obtain solution
Homogeneous DE
It is homogeneous if M(x,y)dx + N(x,y)dy = 0 has all terms of equal degree
to solve:
let:
y = vx
dy = vdx + xdv
and rearrange to form variable separable
Exact DE
M(x,y)dx + N(x,y)dy = 0
is exact if:
dM/dy = dN/dx
to solve:
integrate M wrt x and N wrt y and combine unique terms
Linear DE
dy/dx + y P(x) = Q(x)
General Solution:
y(e^∫Pdx) = ∫Q(e^∫Pdx) dx
Bernoulli’s Equation
dy/dx + y P(x) = (y^n)Q(x)
IF = e^(1-n)∫Pdx
General Solution:
(y^(1-n)) (e^∫(1-n)(Pdx)) = ∫(1-n)(Q)(e^∫(1-n)(Pdx)) dx
Population Growth Problems / Population Dynamics
dP/dt = kP
k-const of proportionality
P-population at any given time
Radioactive Decay Problems
dQ/dt = kQ
k-const of proportionality
Q-amount of substance at any given time
Economics: Continuous compound interest
dP/dt = rP
r-nominal rate of interest
P-money present in the account at any time
