Differential Equations

Explore a comprehensive 3-hour 30-minute tutorial on differential equations, covering a wide range of topics and techniques. Begin with separation of variables and integrating factors for solving ordinary differential equations (ODEs). Progress through exact equations, linear models, and applications of linear ODEs. Delve into advanced concepts such as series solutions, regular singular points, and the method of Frobenius. Study the Legendre equation and its properties. Master the Laplace transform, including its definition, linearity properties, and inverse transformations. Learn to solve initial value problems using Laplace transforms and unit step functions. Examine systems of linear differential equations with real and complex eigenvalues. Conclude with specialized equation types like Bernoulli and homogeneous differential equations. Gain practical problem-solving skills through numerous examples and full solutions throughout the tutorial.

Separation of Variables :: dy/dx + 2xy^2 = 0 :: y'+2xy^2=0.
Separation of Variables :: yln(x) dx/dy = ((y+1) /x)^2.
ODE:: y' + 2xy = x^3 :: Integrating Factor.
ODE :: y' - x/(x+1)y = x :: Integrating Factor for Linear Equations.
Exact Equations :: (1-3/y + x) dy/dx +y = 3/x-1.
Exact Differential Equation IVP:: ((y^3-t^2)/y^5)dy/dt = -t/(2y^4).
Linear Models:: Applications of Linear ODEs.
Radius of Convergence for Series Solution of a Differential Equation Including Complex Sing Pts.
ODE:: y'' - xy' + 2y=0 :: Power Series Solution about an Ordinary Point.
What are Regular Singular Points of Differential Equations?? With 3 Full Examples.
ODE :: xy'' + y' +2xy = 0 :: Method of Frobenius Series Solution about a Regular Singular Point.
Show sin(t) y'' + cos(t) y' +n(n+1) sin(t) y = 0 is a Legendre Equation.
The Definition of the Laplace Transform and Three Basic Examples.
Laplace Transform of a Piecewise Function Using the Definition.
Find the Laplace Transform of cos(kt) using the Definition.
Linearity Property of the Laplace Transform and 7 Useful Transforms to Know! Full Example..
Inverse Laplace Transform and Linearity of Inverse Laplace :: With Examples.
Shifting Laplace Transforms :: The First Translation Theorem for Laplace Transforms.
Inverse Laplace Transform :: Completing the Square :: First Translation Theorem in Reverse.
Inverse Laplace Partial Fraction Decomposition :: Overall Strategy.
Use Laplace Transform to Solve Initial Value Problem :: Full Example with Partial Fraction Decomp.
Laplace Transform With Unit Step Functions :: Second Shifting/Translation Theorem.
Proof of the Convolution Theorem :: Laplace Transforms.
Homogeneous System of Linear Differential Equations :: Real Distinct Eigenvalues.
Homogeneous System of Differential Equations :: Complex Eigenvalues.
Bernoulli DIfferential Equation || xy' -(1+x)y = xy^2.
(x^2+2y^2) dx/dy = xy || Homogeneous Substitution.