Differential Equations

Chapter 6

Differential Equations

Just like how we can have normal equations like f(x), we can also have differential equations like f′(x). Differential equations come in families, and slope fields are the visual representation of a whole family of differential equations on a graph. In this sections, we will look at differential equations and methods to find their solutions.

I hope the music helps! 😁😉

Euler's Method:

Given: y′ = F(x, y), (x-^0, y-^0)

Slope through (x-^0, y-^0):

F(x-^0, y-^0)

Do this: Using a small step h, move along the tangent line until you arrive at the point (x-^1, y-^1)

x-^1

= x-^0 + h

y-^1

= y-^0 + hF(x-^0, y-^0)

Repeat the process for (x-^2, y-^2) and so on…

x-^n

= x-^(n - 1) + h

y-^n

= y-^0 + hF(x-^(n - 1), y-^(n - 1) )

NOTE: You can obtain better approximations of the exact solution by choosing smaller and smaller step sizes.

Chapter 6.2

Differential Equations: Growth and Decay

Theorem 6.1 - Exponential Growth and Decay Model:

If y is a differentiable function of t such that y > 0 and y′ = ky for some constant k, then

y = Ce^kt

C - initial value of y
k - proportionality constant

Exponential growth occurs when k > 0 and exponential decay occurs when k < 0.

To solve for C, use the point (0, y(0))
To solve for k use the value of C previously obtained and any other point on the graph
If not provided with (0, y(0)):
- Find C in terms of t with a point on the graph
- Substitute this value along with another point on the graph into the equation
- Solve for k
- Plug everything back into the equation
- Solve for C

Chapter 6.1

Slope Fields and Euler's Method

y′ = F(x, y) - differential equation (equation for the derivative of y)

F(x-^n, y-^n) - the value obtained if you plug in x-^n and y-^n

A function y = f(x) is a solution of a differential equation if the equation is satisfied when y and its derivatives are replaced by f(x) and its derivatives.

y = e-^(-2x) is a solution of y′ + 2y = 0

general solution - when the constant C is used

Example: y = Ce-^(-2x)

singular solution - when the constant C is replaced by a value

order of a differential equation - determined by the highest-order derivative in the equation

Example: y′ = 4y --> 1st order differential equation

A differential equation of order n has a general solution with n arbitrary constants.

The general solution of a first-order differential equation represents a family of curves known as solution curves.

initial conditions - give the value of the dependent variable or one of its derivatives for a particular value of the independent variable

If you draw a short line segment with slope F(x, y) at selected points (x, y) in the domain of F, then these line segments form a slope field or direction field for the differential equation y′ = F(x, y).
- Each line segment has the slope of the solution curve through that point.
- A slope field shows the general shape of all the solutions.

Chapter 6.3

Separation of Variables and the Logistic Equation

Separation of Variables:

Consider a differential equation that can be written in the form

M(x) + N(y) (dy/dx) = 0

where M is a continuous function of x alone and N is a continuous function of y alone.

Collect all x-terms with dx on one side of the equation
Collect all y-terms with dy on the other side of the equation
Integrate both sides of the equation
Solve for a solution

Such equations are said to be separable, thus the method is called separation of variables.

Some differentiable equations that are not separable in x and y can be made separable by a change in variables. This is true for differential equations in the form y′ = f(x, y), where f is a homogenous function. The function given by f(x, y) is homogenous of degree n if

f(x, y) = t^n * f(x, y)

where n is a real number.

Definition of Homogenous Differential Equation:

A homogeneous differential equation is an equation of the form

M(x, y)dx + N(x, y)dy = 0

where M and N are homogenous functions of the same degree

Theorem 6.2 - Change of Variables for Homogenous Equations:

If M(x, y)dx + N(x, y)dy = 0 is homogenous, then it can be transformed into a differential equation whose variables are separable by the substitution

y = vx

where v is a differentiable function of x.

Consider this:

x^2 + y^2 = C (family of circles)

y = kx (family of lines)

Each circle intersects a line at a right angle.

The two families are said to be mutually orthogonal because each curve in one of the families is called an orthogonal trajectory of the other family.

Logistic Differential Equation:

dy/dt = ky (1 - y/L)

L - carrying capacity
L and k are positive constants.

If 0 < y < L, then dy/dt > 0, and the population increases.

If y > L, then dy/dt < 0, and the population decreases.

Solution of the Logistic Differential Equation:

y = L/(1 + be^-kt)

Chapter 6.4

First-Order Linear Differential Equation

Definition of First-Order Linear Differential Equation:

A first-order linear differential equation is an equation of the form

dy/dx + P(x) y = Q(x)

where P and Q are continuous functions of x. This first-order linear differential equation is said to be in standard form.

Theorem 6.3 - Solution of a First-Order Linear Differential Equation:

An integrating factor for the first-order linear differential equation

y′ + P(x)y = Q(x)

is u(x) = e^∫P(x)dx. The solution of the differential equation is

ye^∫P(x)dx

= ∫Q(x) (e^∫P(x)dx) dx + C

y = (1/u(x)) ∫Q(x) u(x)dx

Bernoulli Equation:

The Bernoulli equation is nonlinear but can be reduced to a linear equation.

y′ + P(x)y = Q(x) y^n

linear if n = 0
separable variables if n = 1

General Solution of the Bernoulli Equation:

y^(1 - n) * e^∫(1 - n) P(x)dx

= ∫(1 - n) Q(x) (e^∫(1 - n) P(x)dx) dx + C