APPM 2360 (Differential Equations and Linear Algebra)

What is an isocline?

A curve along which a function, especially the slope of a differential equation, has a constant value.

What is an equilibrium solution in differential equations?

A constant solution where the rate of change of the dependent variable is zero, meaning the solution does not change over time

What makes an equilibrium solution stable, unstable, and partially stable?

If surrounding slope points on both sides are approaching the solution it is stable. If only one side is approaching than the solution is semi-stable, and if both are moving away than the equilibrium solution is unstable.

If a differential equation cannot be solved and you are using qualitative analysis: what should you be looking into?

Behavior as t → infinity, equilibria, asymptotes, and concavity (sign of y”).

What makes a differential equation non-separable?

When its variables, typically x and y, cannot be isolated on opposite sides of the equation to form f(y) dy = g(x) dx. Usually addition and trig are signs that they aren’t separable.

What is an implicit solution?

A solved differential equation where y is not isolated to one side.

What are the general steps of separation of variables?

• Separate t and y to different sides

• Integrate

• Solve for y (if possible)

What formula for Euler’s Method?

f_n=f_n-1+h*f(t_n-1,y_n-1)

Is this IVP solvable? y’=1/t, y(0)=0. Why or why not?

It is not as y’=ln(t)+C and ln(0) is undefined thus the IVP has no solution.

What is Piccard’s Theorem?

• If f(t,y) is continuous around (t₀,y₀) then there is a solution to the IVP in some range (t₀-h,t₀+h) for some h>0.

• If ∂f/∂y is continuous around (t₀,y₀) than the solution is unique.

What makes a DE linear?

if its dependent variable (y) and its derivatives appear only to the first power, are not multiplied together, and are not within transcendental functions like sine, logarithm, or exponential functions.

What makes a higher order DE homogenous?

If f(x) = 0, meaning there is no non-zero term independent of the dependent variable and its derivatives.

How can you tell if a 1st order DE is homogenous?

• A common method is to substitute 'ty' for 'y' and 'tx' for 'x' into the function f(x, y).

• If f(tx, ty) = tⁿ f(x, y)

  for some power 'n' (where n is the degree), the fu

What is the superposition principle in DE?

In the context of differential equations (DEs), the superposition principle states that if y₁ and y₂ are solutions to a linear homogeneous DE, then any linear combination of these solutions, C₁y₁ + C₂Y₂, where C₁ and C₂ are constants, is also a solution to that same DE.

How do you do the method of undetermined coefficients?

• Consider the associated homogeneous differential equation, where the nonhomogeneous term (e.g., f(x) or G(x)) is set to zero. Then solve for the complimentary function y_h.

• Find the particular solution y_pMake a "judicious guess" for the form of y_p based on f(x). For example:

  • If f(x) is a polynomial, guess a polynomial of the same degree.

  • If f(x) is a combination of sine and cosine, guess a similar combination.

• Substitute this guessed y_p into the original nonhomogeneous differential equation.

• Determine the coefficients in your guess by solving the resulting equations.

• Combine the homogeneous and particular solutions to get the total solution: y = y_h + y_p. 

How do you do the integrating factor method?

• Standard Form:

  First, rewrite the linear first-order differential equation in the standard form: 

  • dy/dx + p(x)y = q(x) 

• Calculate the Integrating Factor:

  The integrating factor, μ(x), is found by taking the exponential of the integral of p(x): 

  • μ(x) = e^∫p(x)dx 

• Multiply by the Integrating Factor:

  Multiply the entire standard-form equation by μ(x). This step is crucial because it makes the left-hand side a perfect derivative of a product: 

  • μ(x)(dy/dx + p(x)y) = μ(x)q(x) 

  • d/dx(μ(x)y) = μ(x)q(x) 

• Integrate both sides

• Solve for y.

What is a transient state solution?

It's the part of the total solution that describes the system's behavior as it changes from its initial state to a stable state.

What is a steady state?

This is the long-term behavior of the system once it has settled into a stable condition, where changes become minimal. 

What is the relationship between the transient and steady state solutions?

• The total solution to a dynamic system is often the sum of its transient and steady-state parts.

• As time goes on, the transient part of the solution diminishes, and the system's behavior becomes dominated by the steady-state solution.

What are the two properties of a linear operator L(y)

• Additivity, meaning L(u + v) = L(u) + L(v)

• Homogeneity of degree 1, meaning L(cu) = cL(u) for all functions u and constants c

When does the superposition principle apply?

When the equation is linear and homogenous.

Isoclines of autonomous first order differential equations will always be what?

Horizontal lines

What makes a DE autonomous?

If the independent variable (usually t) does not appear in the equation.

What makes a matrix consistent or inconsistent?

Consistent if it has at least one solution (a unique solution or infinitely many solutions) and inconsistent if it has no solution

What are elementary row solutions?

Consistent if it has at least one solution (a unique solution or infinitely many solutions) and inconsistent if it has no solution

What is reduced Row Echelon Form?

• it is in row echelon form

• every leading entry (pivot) in each non-zero row is a 1

• every column containing a leading 1 has zeros in all its other entries

What is row echelon form?

• all zero rows are at the bottom of the matrix

• the first non-zero entry (called a "pivot" or "leading entry") of any non-zero row is to the right of the pivot in the row above it

• all entries in a column below a pivot are zero