Cheat Sheet
Table of Contents Top
First Order
Linear
Separation of Variables
Exact
Homogeneous Linear substitution)
Bernoulli
Separation of Variables
Mathematical Models
Radioactive Decay
Newton's Law of Cooling
Spread of Disease
Mixture
Tank Draining
Series Circuit
Falling Bodies
Falling Bodies and Air Resistance
Suspended Cables
Growth and Decay
Logistic Equation
Chemical Reactions
Second Order
Constant Coefficients
Case 1 Distinct Real Roots
Case 2 Repeated Real Roots
Case 3 Conjugate Complex Roots
Undetermined Coefficients
Variation of Parameters
Cauchy-Euler
Case 1 Distinct Real Roots
Case 2 Repeated Real Roots
Case 3 Conjugated Complex Roots
Non-Linear
Linear Models
Hooke's Law
Newton's Second Law
Free Undamped Motion
Double Spring System
Cheat Sheet 1
, Free Damped Motion
Case 1 Overdamped
Case 2 Critically damped
Case 3 Underdamped
Alternative Form of x(t)
Differential Equation of Driven Motion With Damping
LRCSeries Circuit
Kirchhoff's Second Law
Non-Linear Models
Springs
Hard and Soft Springs
Nonlinear Pendulum
Telephone Wires
Rocket Motion
Variable Mass
Power Series
Linear Systems
Matrix Form of a Linear System
Superposition Principle
General Solution - Homogeneous Systems
General Solution - Nonhomogeneous Systems
Eigenvalues and Eigenvectors
Distinct Real Eigenvalues
Repeated Eigenvalues
Multiplicity Two
Multiplicity Three
Complex Eigenvalues
Undetermined Coefficients
Variation of Parameters
First Order
Linear
dy
+ P (t)y = g(t)
dx
μ(t) = e∫ (p(t)dt
μ(t)P (t) = ∫ μ(t)g(t)dt
Cheat Sheet 2
, Separation of Variables
dy
= g(x)h(y)
dx
dy
p(y) = g(x)
dx
∫ p(y)dy = ∫ g(x)dx
Exact
M(x, y)dx + N(x, y)dy = 0
∂M ∂N
=
∂y ∂x
∂f ∂F
M(x, y) = , N(x, y) =
∂x ∂y
∂f ∂
= ∫ M(x, y)dx + g′ (y) = N(x, y)
∂y ∂y
∂
g′ (y) = N(x, y) − ∫ M(x, y)dx
∂y
∂ ∂
[N(x, y) − ∫ M(x, y)dx]
∂x ∂y
∂N ∂ ∂
− ( ∫ M(x, y)dx)
∂x ∂y ∂x
∂N ∂M
= =0
∂x ∂y
f(x, y) = ∫ N(x, y)dy + h(x)
∂
h′ (x) = M(x, y) − ∫ N(x, y)dy
∂x
Non-exact differential equations
μ(x, y)
μ(x, y)M(x, y)dx + μ(x, y)N(x, y)dy = 0
M y −Nx
μ(x) = e∫ N
dx
Nx −M y
μ(x) = e∫ M
dy
Cheat Sheet 3
Table of Contents Top
First Order
Linear
Separation of Variables
Exact
Homogeneous Linear substitution)
Bernoulli
Separation of Variables
Mathematical Models
Radioactive Decay
Newton's Law of Cooling
Spread of Disease
Mixture
Tank Draining
Series Circuit
Falling Bodies
Falling Bodies and Air Resistance
Suspended Cables
Growth and Decay
Logistic Equation
Chemical Reactions
Second Order
Constant Coefficients
Case 1 Distinct Real Roots
Case 2 Repeated Real Roots
Case 3 Conjugate Complex Roots
Undetermined Coefficients
Variation of Parameters
Cauchy-Euler
Case 1 Distinct Real Roots
Case 2 Repeated Real Roots
Case 3 Conjugated Complex Roots
Non-Linear
Linear Models
Hooke's Law
Newton's Second Law
Free Undamped Motion
Double Spring System
Cheat Sheet 1
, Free Damped Motion
Case 1 Overdamped
Case 2 Critically damped
Case 3 Underdamped
Alternative Form of x(t)
Differential Equation of Driven Motion With Damping
LRCSeries Circuit
Kirchhoff's Second Law
Non-Linear Models
Springs
Hard and Soft Springs
Nonlinear Pendulum
Telephone Wires
Rocket Motion
Variable Mass
Power Series
Linear Systems
Matrix Form of a Linear System
Superposition Principle
General Solution - Homogeneous Systems
General Solution - Nonhomogeneous Systems
Eigenvalues and Eigenvectors
Distinct Real Eigenvalues
Repeated Eigenvalues
Multiplicity Two
Multiplicity Three
Complex Eigenvalues
Undetermined Coefficients
Variation of Parameters
First Order
Linear
dy
+ P (t)y = g(t)
dx
μ(t) = e∫ (p(t)dt
μ(t)P (t) = ∫ μ(t)g(t)dt
Cheat Sheet 2
, Separation of Variables
dy
= g(x)h(y)
dx
dy
p(y) = g(x)
dx
∫ p(y)dy = ∫ g(x)dx
Exact
M(x, y)dx + N(x, y)dy = 0
∂M ∂N
=
∂y ∂x
∂f ∂F
M(x, y) = , N(x, y) =
∂x ∂y
∂f ∂
= ∫ M(x, y)dx + g′ (y) = N(x, y)
∂y ∂y
∂
g′ (y) = N(x, y) − ∫ M(x, y)dx
∂y
∂ ∂
[N(x, y) − ∫ M(x, y)dx]
∂x ∂y
∂N ∂ ∂
− ( ∫ M(x, y)dx)
∂x ∂y ∂x
∂N ∂M
= =0
∂x ∂y
f(x, y) = ∫ N(x, y)dy + h(x)
∂
h′ (x) = M(x, y) − ∫ N(x, y)dy
∂x
Non-exact differential equations
μ(x, y)
μ(x, y)M(x, y)dx + μ(x, y)N(x, y)dy = 0
M y −Nx
μ(x) = e∫ N
dx
Nx −M y
μ(x) = e∫ M
dy
Cheat Sheet 3
