Differential Equations Exam 1

Differential Equation

An equation involving derivatives of a function, representing a relationship between the function and its rates of change.

Ordinary

differential equation involving a single independent variable.

Partial

differential equation involving multiple independent variables.

Order

the highest derivative present in a differential equation.

Solution

a function that satisfies a differential equation.

Partial Derivative

the derivative of a function with respect to one variable while keeping others constant.

Ordinary DE

differential equations that contain functions of one independent variable and their derivatives.

Linear DE

a type of differential equation where the dependent variable and its derivatives appear linearly.

Homogeneous

all its terms are a function of the dependent variable and its derivatives, and it equals zero.

Non-homogeneous

a differential equation that includes terms that are not solely functions of the dependent variable and its derivatives, typically including a non-zero function.

Initial Value Problem

a type of differential equation that specifies the value of the dependent variable at a particular point, allowing for a unique solution.

Initial Conditions

the specific values of the dependent variable and its derivatives at a given point, used to solve initial value problems.

First Order DE

a differential equation involving only the first derivative of the dependent variable.

Direction Field

a graphical representation of the slopes of solutions to a first-order differential equation at given points in the plane. It helps visualize the behavior of solutions without solving the equation explicitly.

Trajectory

the path that a solution of a differential equation follows in the phase plane, representing the evolution of the system over time.

Isocline

a curve in the direction field where the slope of the solution is constant, helping to visualize the behavior of solutions to a differential equation.

Autonomous

No X on the right, equal to 0

Separable

A type of differential equation that can be expressed as the product of a function of the dependent variable and a function of the independent variable, allowing for separation of variables to solve.

Logistic

A type of differential equation that models population growth, characterized by an S-shaped curve and includes parameters for carrying capacity and growth rate. y’=ky-ay²

Exact

A type of differential equation that can be solved by finding an integrating factor, making it possible to express the equation in a form that allows direct integration. M(x,y)dx+N(x,y)dy=0

Closed

A type of differential equation where the solution set forms a closed curve in the phase plane, indicating that the system is bounded and does not diverge to infinity. dM/dy=dN/dx

Closed theorem

A closed differential in a rectangle R in the xy-plane is exact in R.

First-Order Existence and Uniqueness Theorem

States that if a function satisfies certain conditions, then there exists a unique solution to a first-order differential equation near a given point.

Numerical Solution by Euler’s Method

Tangent line approximations to find a solution. y_k+1=y_k+hf(x_k,y_k)

Basic Integration

y’=f(x)…y=integral of xdx

Separable

y’=p(x)/q(y)…integral q(y)dy = integral p(x)dx

Exponential

y’=ky…integral dy/y = integral kdx…y=y₀e^kx

Logistic

y’=ky-Ay²…integral dy/(ky-Ay²) = integral dx…y=(k/A)/(1+Ce^-kx) for some constant C

First-Order Linear

y’+p(x)y=q(x)…multiplier u(x)=e^{integral p(x)dx}…y=(1/u(x)) * integral (u(x)q(x)dx)

Exact

M(x,y)dx+N(x,y)dy=0…integrate Mdx=F₀(x,y)+C(y)…differentiate partial F₀/y + C’(y) and set equal to N. Simplifies to C’(y)=some function of y. All x’s need to cancel out. Integrate to find C(y) and the solutions are F(x,y)=F₀(x,y)+C(y)=C for all constants C

First-Order Linear Existence and Uniqueness Theorem

For y’+p(x)y=q(x), if p and q are continuous in a rectangle containing (x₀,y₀), the IVP has a unique solution valid in the whole rectangle

Mixing Problems

x’(t)=inflow-outflow

Population Model by Logistic Equation

p’(t)=kp-Ap². Solution: y=p₁/(1+Ce^-kt) where p₁=k/A is the long-term equilibrium solution

Falling Bodies with Air Resistance (k>0)

mdv/dt=mg-kv, v(0)=v₀. Solution: v=mg/k + (v₀-mg/k)e^-kt/m

Falling Bodies with Air Resistance (x(0)=0)

x’(t)=v. Solution: x=mgt/k + m(v₀-mg/k)/k * (1-e^-kt/m)

Second-Order Linear Equations with Constant Coefficients

ay”+by’+cy=0 with both auxiliary roots being real

Auxiliary Equation

ar²+br+c=0

Auxiliary Roots

a(r-r₁)(r-r₂)

General Solution of Real & Distinct Roots

y=c₁e^r₁x+c₂e^r₂x

General Solution of Roots Equal to r

y=c₁e^rx+c₂xe^rx