Differential Equations Study Notes
Introduction to Differential Equations
Definition of Differential Equations (D.E.): An equation that contains the derivative of an unknown function, which is the dependent variable.
Example 1: A first-order differential equation can be expressed as y'(t) where:
t: independent variable
y: dependent variable
Example 2: A second-order differential equation can be given as y''(t) + 2y(t) + heta(t) = 0.
In this case, y(t) represents the unknown function, while the terms include its second-order derivative and other functions.
Newton's Second Law: Expressed as F = ma, rephrased to v'(t) = x''(t) = - heta(t) = m x''(t).
Variables:
x(t): position
u(t): velocity
a(t): acceleration
Solving Differential Equations
Methodology to Solve D.E.:
Use the provided function to compute necessary derivatives.
Substitute back into the equation and simplify.
Confirm that the left-hand side (L.H.S) equals the right-hand side (R.H.S).
Example 3: Verify whether y(t) = e^{t} - Y'(t) e^{t} + zy = 2e^{2} + 2e^{t} - 4e^{t} is a solution to the differential equation defined as:
Results in Y''_{4}Y = 0 where R.H.S = 0.
Classification of Differential Equations
Ordinary Differential Equations (ODE): The unknown function depend on only one independent variable.
Example: f(t), often referred to as a simple ODE.
Partial Differential Equations (PDE): The unknown function depends on more than one independent variable.
Example: f(t, s) is a PDE where partial derivatives exist.
Continuing Classification:
The order of the D.E. is determined by the highest degree derivative present in the equation.
Linear vs Non-linear:
Linear D.E.: All terms involving the unknown function and its derivatives occur to the first power, with coefficients dependent only on the independent variable.
Non-linear D.E.: The terms violate linearity conditions (i.e., powers greater than 1).
General Forms of Differential Equations
General Form of Linear ODE:
a{n}(x) y^{(n)} + a{n-1}(x) y^{(n-1)} + … + a1(x) y' + a0(x) y = F(x)Where:
y^{(n)}: nth derivative of y
F(x): a function of the independent variable
Determining Linear vs Non-linear:
Example a: y'(t) + y^2 e^0 is non-linear.
Example b: f(x) + x f'(x) = 2 is linear.
Explicit vs Implicit Solution:
Explicit: y = g(x) that satisfies the differential equation for all x in the interval considered.
Implicit: F(x, y, y', …, y^{(n)}) = 0 involves mixing the dependent variable with its derivatives, proving more complex to isolate solutions.
Examples of Solving D.E.
Example: Verify if y = 3 ext{sin}(2x) + e^{-x} is a solution:
Compute derivatives:
y' = 6 ext{cos}(2x) - e^{-x}
y'' = -12 ext{sin}(2x) - e^{-x}Check against the defined D.E.
Example 2: Determine whether heta(x) = e^{x} - x is a solution to the equation relating to L.H.S = R.H.S.
Initial Value Problems (IVP)
Definition: An initial value problem is a D.E. that includes additional initial conditions, such as y(0) = y_0.
Example of IVP: Solve for y given that the general solution is y(x) = C1 e^x + C2 e^{-2x}. Determine constants using conditions like y(0) = 2 and y'(0) = 1.
Direction Fields and Graphical Approach
Definition of Direction Field: A visual representation that indicates the slope of solutions to a D.E. across a certain plane.
Example: For dy/dx = x - 4, the direction at any point (x,y) on the plane can be plotted to visualize the behavior of the solution curves.
Use slope strategies to sketch and utilize for initial value problems.
Autonomous ODE
Definition: An autonomous ODE is one that is written as dy/dx = f(y) and does not contain the independent variable explicitly.
Example: In the population model dP/dt = -2P(P - 4), where P changes with time t, it indicates how population dynamics can be depicted without directly depending on time.