Differential Equations | Act I

ordinary differential equation (ODE)

derivatives WRT only ONE independent variable

partial differential equation (PDE)

several derivatives WRT >1 independent variable

• ∇² (Laplacian operator) indicates PDE

• usually indicated by ∂x (partial diff) if x is a function that has more than 1 variable as inputs

order

the number of the highest derivative in a function

• order can be measured in both ODE & PDE

linear

the unknown variable & its derivatives appear

• at power 0 or 1

• not multiplied by each other, but maybe multiplied with another function x

• not inside nonlinear functions (functions whose Taylor expansion has higher powers, like sin)

can be ODE or PDE

linear homogeneous

all terms include the unknown variable; = 0

linear inhomogeneous

not all terms include the unknown; = constant or other function

Solution to y’ = ay

y = Ae^ax

Solution to y’ + a²y = 0

y = Ae^ax + Be^-ax

y’ - a²y = 0

y = A cos(ax) + B sin(ax)

explicit solution

unknown function is directly in terms of independent variables

y = f(x)

implicit solution

unknown function is not directly in terms of independent variables

F(x,y) = 0

general vs specific solution

general

• all solutions for differential equation

• usually contains arbitrary constants

specific

• follows initial conditions

• no arbitrary unknown constants

integrating factor method

for first order linear ODEs of the form:

y’ + py = q

multiply by u = e^(int p dt)

reverse product rule

technique for solving 1st order linear ODEs when one side of the equation is the derivative of a product

variation of parameters

technique for solving inhomogeneous 1st order ODEs when homogeneous solution is easier to solve

1. convert to homogeneous form

2. compute y and y’ in terms of f(x) which takes C’s place

3. substitute y and y’ into og inhomogeneous equation

4. integrate both sides WRT s, usually f(x) is isolateable by now

5. substitute f(x) back into y_h(x)

singular solution

solution to a differential equation that cannot be obtained by plugging in values of constants from the general solution — but still satisfies the equation

exact equation

equation that can be written in the form of

dψ = Mdx + Ndy = 0

Solving an exact equation steps

1. Check My = Nx

2. Integrate M WRT x and add h(y)

3. Differentiate WRT y

4. Set equal to N(x,y)

5. Integrate h’(y)

6. Substitute h(y) back into original equation for potential function

Solving exact equation when M & N are not exact

1. Multiply Mdx + Ndy = 0 by μ

2. My = μ’M + μM’ WRT y

3. Nx = μ’N + μN’ WRT x

4. Set My = Nx

5. Assume μ = μ(x), and thus set μy = 0

6. Find μ(x), then multiply the original equation by μ(x) to make it exact

Substitution when dy/dx = F(y/x)

1. Use v(x) = y/x, y = xv, y’ = v + xvb’

2. Separate new equation by term

3. Integrate both sides

4. Substitute y back in for v

5. Isolate y